Recent zbMATH articles in MSC 35Lhttps://zbmath.org/atom/cc/35L2024-07-17T13:47:05.169476ZWerkzeugOn the joint evolution problem for a scalar field and its singularityhttps://zbmath.org/1536.350132024-07-17T13:47:05.169476Z"Agashe, Aditya"https://zbmath.org/authors/?q=ai:agashe.aditya"Lee, Ethan"https://zbmath.org/authors/?q=ai:lee.ethan-simpson"Tahvildar-Zadeh, Shadi"https://zbmath.org/authors/?q=ai:tahvildar-zadeh.shadiSummary: In the classical electrodynamics of point charges in vacuum, the electromagnetic field, and therefore the Lorentz force, is ill-defined at the locations of the charges. Kiessling resolved this problem by using the momentum balance between the field and the particles, extracting an equation for the force that is well-defined where the charges are located, so long as the field momentum density is locally integrable in a neighborhood of the charges.
We examine the effects of such a force by analyzing a simplified model in one space dimension. We study the joint evolution of a massless scalar field together with its singularity, which we identify with the trajectory of a particle. The static solution arises in the presence of no incoming radiation, in which case the particle remains at rest forever. We will prove the stability of the static solution for particles with positive bare mass by showing that a pulse of incoming radiation that is compactly supported away from the point charge will result in the particle eventually coming back to rest. We will also prove the nonlinear instability of the static solution for particles with negative bare mass by showing that an incoming radiation with arbitrarily small amplitude will cause the particle to reach the speed of light in finite time. We conclude by discussing modifications to this simple model that could make it more realistic.Invariance analysis and closed-form solutions for the beam equation in Timoshenko modelhttps://zbmath.org/1536.350212024-07-17T13:47:05.169476Z"Al-Omari, S. M."https://zbmath.org/authors/?q=ai:al-omari.s-m"Hussain, A."https://zbmath.org/authors/?q=ai:hussain.akhtar"Usman, M."https://zbmath.org/authors/?q=ai:usman.mustofa|usman.murat|usman.muhammad|usman.mohammad|usman.mahamood|usman.muhammad-rashid|usman.muhammad.1"Zaman, F. D."https://zbmath.org/authors/?q=ai:zaman.fiazud-dinSummary: Our research focuses on a fourth-order partial differential equation (PDE) that arises from the Timoshenko model for beams. This PDE pertains to situations where the elastic moduli remain constant and an external load, represented as F, is applied. We thoroughly analyze Lie symmetries and categorize the various types of applied forces. Initially, the principal Lie algebra is two-dimensional, but in certain noteworthy cases, it extends to three dimensions or even more. For each specific case, we derive the optimal system, which serves as a foundation for symmetry reductions, transforming the original PDE into ordinary differential equations. In certain instances, we successfully identify exact solutions using this reduction process. Additionally, we delve into the conservation laws using a direct method proposed by Anco, with a particular focus on specific classes within the equation. The findings we have presented in our study are indeed original and innovative. This study serves as compelling evidence for the robustness and efficacy of the Lie symmetry method, showcasing its ability to provide valuable insights and solutions in the realm of mathematical analysis.Asymptotics for singular limits via phase functionshttps://zbmath.org/1536.350262024-07-17T13:47:05.169476Z"Nordmann, Samuel"https://zbmath.org/authors/?q=ai:nordmann.samuel"Schochet, Steve"https://zbmath.org/authors/?q=ai:schochet.steven-hSummary: The asymptotic behavior of solutions as a small parameter tends to zero is determined for a variety of singular-limit PDEs. In some cases even existence for a time independent of the small parameter was not known previously. New examples for which uniform existence does not hold are also presented. Our methods include both an adaptation of geometric optics phase analysis to singular limits and an extension of that analysis in which the characteristic variety determinant condition is supplemented with a periodicity condition.Numerical upscaling via the wave equation with perfectly matched layershttps://zbmath.org/1536.350302024-07-17T13:47:05.169476Z"Arjmand, Doghonay"https://zbmath.org/authors/?q=ai:arjmand.doghonaySummary: One of the main ingredients of existing multiscale numerical methods for homogenization problems is an accurate description of the coarse scale quantities, e.g., the homogenized coefficient via local microscopic computations. Typical multiscale frameworks use local problems that suffer from the so-called resonance or cell-boundary error, dominating the all other errors in multiscale computations. Previously, the second order wave equation was used as a local problem to eliminate such an error. Although this approach eliminates the resonance error totally, the computational cost of the method is known to increase with increasing wave speed. In this paper, the possibility of integrating perfectly matched layers to the local wave equation is explored. In particular, questions in relation with accuracy and reduced computational costs are addressed. Numerical simulations are provided in a simplified one-dimensional setting to illustrate the ideas.
For the entire collection see [Zbl 1515.60023].A spectral ansatz for the long-time homogenization of the wave equationhttps://zbmath.org/1536.350322024-07-17T13:47:05.169476Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoine"Ruf, Matthias"https://zbmath.org/authors/?q=ai:ruf.matthiasSummary: Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present work is a new ansatz for the long-time two-scale expansion inspired by spectral analysis. Based on this spectral ansatz, we extend and refine all previous results in the field, proving homogenization up to optimal timescales with optimal error estimates, and covering all the standard assumptions on heterogeneities (both periodic and stationary random settings).Optimal control problem governed by wave equation in an oscillating domain and homogenizationhttps://zbmath.org/1536.350332024-07-17T13:47:05.169476Z"Faella, Luisa"https://zbmath.org/authors/?q=ai:faella.luisa"Raj, Ritu"https://zbmath.org/authors/?q=ai:raj.ritu"Sardar, Bidhan Chandra"https://zbmath.org/authors/?q=ai:sardar.bidhan-chandraSummary: In this article, we consider the optimal control problem governed by the wave equation in a 2-dimensional domain \(\Omega_{\epsilon}\) in which the state equation and the cost functional involves highly oscillating periodic coefficients \(A^{\epsilon}\) and \(B^{\epsilon}\), respectively. This paper aims to examine the limiting behavior of optimal control and state and identify the limit optimal control problem, which involves the influences of the oscillating coefficients.On orbital stability of solitons for 2D Maxwell-Lorentz equationshttps://zbmath.org/1536.350482024-07-17T13:47:05.169476Z"Komech, Alexander"https://zbmath.org/authors/?q=ai:komech.alexander-ilich"Kopylova, Elena"https://zbmath.org/authors/?q=ai:kopylova.elena-aSummary: We prove the orbital stability of soliton solutions for 2D Maxwell-Lorentz system with extended charged particle. The solitons corresponds to the uniform motion and rotation of the particle. We reduce the corresponding Hamilton system by the canonical transformation via transition to a comoving frame. The solitons are the critical points of the reduced Hamiltonian. The key point of the proof is a lower bound for the Hamiltonian.Laminated Timoshenko beam without complementary dissipationhttps://zbmath.org/1536.350542024-07-17T13:47:05.169476Z"Alves, M. S."https://zbmath.org/authors/?q=ai:alves.margareth-silva"Monteiro, R. N."https://zbmath.org/authors/?q=ai:monteiro.rodrigo-nunesSummary: In this study, the stability problem of a laminated beam with only structural damping is analyzed. The results obtained in this study improve the analysis of the problem by investigating stability without introducing additional dissipation. This is accomplished by considering only the usual assumption of equal wave velocities as the stability criterion.Decay rates of strongly damped Infinite laminated beamshttps://zbmath.org/1536.350562024-07-17T13:47:05.169476Z"Bautista, G. J."https://zbmath.org/authors/?q=ai:bautista.george-j"Cabanillas, V. R."https://zbmath.org/authors/?q=ai:cabanillas.victor-r"Potenciano-Machado, L."https://zbmath.org/authors/?q=ai:potenciano-machado.leyter"Méndez, T. Quispe"https://zbmath.org/authors/?q=ai:mendez.teofanes-quispeSummary: In this paper, we study the stability of a Timoshenko laminated beam model with Kelvin-Voigt dampings. We consider both the case of the fully damped and partially damped system in which two dampings are effective on the system. Using the energy method, Fourier analysis and the construction of functionals with suitable weights, we obtain exponential and polynomial decay estimates for the solution of the system and its higher-order derivatives. The polynomial decay rates obtained depend on the regularity of the initial data and vary according to the position of the damping terms.Wave equations with a damping term degenerating near low and high frequency regionshttps://zbmath.org/1536.350572024-07-17T13:47:05.169476Z"Charão, Ruy Coimbra"https://zbmath.org/authors/?q=ai:charao.ruy-coimbra"Ikehata, Ryo"https://zbmath.org/authors/?q=ai:ikehata.ryoSummary: We consider wave equations with a nonlocal polynomial type of damping depending on a small parameter \(\theta \in (0,1)\). This research is a trial to consider a new type of dissipation mechanisms produced by a bounded linear operator for wave equations. These researches were initiated in a series of our previous works with various dissipations modeled by a logarithmic function published in
[\textit{R. C. Charão} et al., Math. Methods Appl. Sci. 44, No. 18, 14003--14024 (2021; Zbl 1479.35089);
\textit{R. C. Charão} and \textit{R. Ikehata}, Z. Angew. Math. Phys. 71, No. 5, Paper No. 148, 26 p. (2020; Zbl 1447.35051);
\textit{A. Piske} et al., J. Differ. Equations 311, 188--228 (2022; Zbl 1481.35067)].
The model of dissipation considered in this work is probably the first defined by more than one sentence and it opens field to consider other more general. We obtain an asymptotic profile and optimal estimates in time of solutions as \(t \rightarrow \infty\) in \(L^2\)-sense, particularly, to the case \(0<\theta <1/ 2\).Asymptotic stabilization for Bresse transmission systems with fractional dampinghttps://zbmath.org/1536.350622024-07-17T13:47:05.169476Z"Hao, Jianghao"https://zbmath.org/authors/?q=ai:hao.jianghao"Wang, Dingkun"https://zbmath.org/authors/?q=ai:wang.dingkunSummary: In this article, we study the asymptotic stability of Bresse transmission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index \(\theta\in [0,1]\) and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with \(\theta\neq 0\), the system is not exponential stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with \(\theta=0\), the system is exponential stable.Existence and general decay of solution for nonlinear viscoelastic two-dimensional beam with a nonlinear delayhttps://zbmath.org/1536.350682024-07-17T13:47:05.169476Z"Lekdim, Billal"https://zbmath.org/authors/?q=ai:lekdim.billal"Khemmoudj, Ammar"https://zbmath.org/authors/?q=ai:khemmoudj.ammarSummary: We investigate the longitudinal and transversal vibrations of the viscoelastic beam with nonlinear tension and nonlinear delay term under the general decay rate for relaxation function. The existence theorem is proved by the Faedo-Galerkin method and using suitable Lyapunov functional to establish the general decay result.Stabilization of the viscoelastic wave equation with variable coefficients and a delay term in nonlocal boundary feedbackhttps://zbmath.org/1536.350712024-07-17T13:47:05.169476Z"Li, Sheng-Jie"https://zbmath.org/authors/?q=ai:li.shengjie"Chai, Shugen"https://zbmath.org/authors/?q=ai:chai.shugenUsing the Faedo-Galerkin approximation, denseness argument, an energy functional and the Riemannian geometry method, the authors present sufficient conditions for existence and uniqueness of strong and weak solutions as well as exponential decay of energy to a class of viscoelastic wave equation with variable coefficients and a delay in nonlinear and nonlocal boundary dissipation.
Reviewer: Jin Liang (Shanghai)Energy decay for wave equations with a potential and a localized dampinghttps://zbmath.org/1536.350722024-07-17T13:47:05.169476Z"Li, Xiaoyan"https://zbmath.org/authors/?q=ai:li.xiaoyan"Ikehata, Ryo"https://zbmath.org/authors/?q=ai:ikehata.ryoSummary: We consider the total energy decay together with the \(L^2\)-bound of the solution itself of the Cauchy problem for wave equations with a short-range potential and a localized damping, where we treat it in the one-dimensional Euclidean space \(\mathbb{R}\). To study these, we adopt a simple multiplier method. In this case, it is essential that compactness of the support of the initial data not be assumed. Since this problem is treated in the whole space, the Poincaré and Hardy inequalities are not available as have been developed for the exterior domain case with \(n \geq 1\). However, the potential is effective for compensating for this lack of useful tools. As an application, the global existence of a small data solution for a semilinear problem is demonstrated.Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weighthttps://zbmath.org/1536.350732024-07-17T13:47:05.169476Z"Makheloufi, Hocine"https://zbmath.org/authors/?q=ai:makheloufi.hocine"Bahlil, Mounir"https://zbmath.org/authors/?q=ai:bahlil.mounirSummary: In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights
\[
u_{tt}(t)+\mathcal{A}u(t)-\int_0^t g (t-s)\mathcal{A} u(s) ds+\mu_1 (t) u_t (t)+ \mu_2 (t) u_t (t-\tau (t)) =0\; \forall t>0,
\]
together with suitable initial conditions. We first prove the existence of a unique global weak solution by means of the classical Faedo-Galerkin method. Then, by assuming the general condition:
\[
g'(t) \le - \xi (t) H(g(t)), \quad\forall t\geq 0,
\]
where \(H\) is a positive increasing and convex function and \(\xi\) is a positive function which is not necessarily monotone, we establish optimal explicit and general stability estimates which rely on the well-known multipliers method and some properties of convex functions. This study generalizes and improves many earlier ones in the existing literature.Exponential stability of a laminated beam system with thermoelasticity of type III and distributed delayhttps://zbmath.org/1536.350742024-07-17T13:47:05.169476Z"Mpungu, Kassimu"https://zbmath.org/authors/?q=ai:mpungu.kassimuSummary: In this work, we consider a one-dimensional laminated Timoshenko beam system with thermoelasticity of type III and distributed delay. Our concern is to investigate the exponential stability of the vibrations in the system without structural damping. By exploiting the perturbed energy method with appropriate assumptions on the delay feedback and speeds of wave propagation, we establish that the unique dissipation through the thermal effect is sufficient for exponential decay of the solution even in the presence of distributed delay.Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy-Ventcel-type boundary conditionshttps://zbmath.org/1536.350782024-07-17T13:47:05.169476Z"Simion Antunes, José G."https://zbmath.org/authors/?q=ai:simion-antunes.jose-g"Cavalcanti, Marcelo M."https://zbmath.org/authors/?q=ai:cavalcanti.marcelo-moreira"Cavalcanti, Valéria N. Domingos"https://zbmath.org/authors/?q=ai:domingos-cavalcanti.valeria-nevesSummary: We study the wellposedness and stabilization for a Cauchy-Ventcel problem in an inhomogeneous medium \(\Omega \subset \mathbb{R}^2\) with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood \(\omega\) of the boundary according to the geometric control condition. We, in particular, generalize substantially the work due to \textit{A. F. Almeida} et al. [Commun. Contemp. Math. 23, No. 3, Article ID 1950072, 38 p. (2021; Zbl 1458.35268)], in what concerns an exponential growth for source term instead of a polynomial one. We give a proof based on the truncation of a equivalent problem and passage to the limit in order to obtain in one shot, the energy identity as well as the observability inequality, which are the essential ingredients to obtain uniform decay rates of the energy. We show that the energy of the equivalent problem goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase space. One advantage of our proof is that the decay rate is independent of the nonlinearity.
{\copyright} 2023 Wiley-VCH GmbH.Sharp polynomial decay for waves damped from the boundary in cylindrical waveguideshttps://zbmath.org/1536.350792024-07-17T13:47:05.169476Z"Wang, Ruoyu P. T."https://zbmath.org/authors/?q=ai:wang.ruoyu-p-tSummary: We study the decay of global energy for the wave equation with Hölder continuous damping placed on the \(C^{1,1}\)-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp \(t^{-1/2}\)-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is \(t^{-1/3}\)-decay when the damping is uniformly bounded from below on the cylindrical wall.Energy decay analysis for porous elastic system with thermoelasticity of type III: a second spectrum approachhttps://zbmath.org/1536.350832024-07-17T13:47:05.169476Z"Zougheib, Hamza"https://zbmath.org/authors/?q=ai:zougheib.hamza"El Arwadi, Toufic"https://zbmath.org/authors/?q=ai:el-arwadi.touficSummary: Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green-Naghdi law of heat conduction into our study. To begin with, we use the Faedo-Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.Uniform attractors of non-autonomous suspension bridge equations with memoryhttps://zbmath.org/1536.350922024-07-17T13:47:05.169476Z"Wang, Lulu"https://zbmath.org/authors/?q=ai:wang.lulu"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhen|ma.qiaozhen.1Summary: In this article, we investigate the long-time dynamical behavior of non-autonomous suspension bridge equations with memory and free boundary conditions. We first establish the well-posedness of the system by means of the maximal monotone operator theory. Secondly, the existence of uniformly bounded absorbing set is obtained. Finally, asymptotic compactness of the process is verified, and then the existence of uniform attractors is proved for non-autonomous suspension bridge equations with memory term.Blow up and lifespan of solutions for elastic membrane equation with delayhttps://zbmath.org/1536.350962024-07-17T13:47:05.169476Z"Benzahi, Mourad"https://zbmath.org/authors/?q=ai:benzahi.mourad"Zaraï, Abderrahmane"https://zbmath.org/authors/?q=ai:zarai.abderrahmane"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Jan, Rashid"https://zbmath.org/authors/?q=ai:jan.rashid"Iqbal, Mujahid"https://zbmath.org/authors/?q=ai:iqbal.mujahidSummary: The primary objective of this research is to examine a nonlinear elastic membrane equation incorporating delay and source terms within a bounded domain. We obtain sufficient conditions on the initial data and the involved functionals for which the energy of solutions with non positive initial energy as well as positive initial energy blow up in a finite-time. In addition, this research work provides estimates for the lifespan of these solutions.On blowup for the supercritical quadratic wave equationhttps://zbmath.org/1536.350982024-07-17T13:47:05.169476Z"Csobo, Elek"https://zbmath.org/authors/?q=ai:csobo.elek"Glogić, Irfan"https://zbmath.org/authors/?q=ai:glogic.irfan"Schörkhuber, Birgit"https://zbmath.org/authors/?q=ai:schorkhuber.birgitSummary: We study singularity formation for the quadratic wave equation in the energy supercritical case, i.e., for \(d \geq 7\). We find in closed form a new, nontrivial, radial, self-similar blow-up solution \(u^*\) which exists for all \(d \geq 7\). For \(d=9\), we study the stability of \(u^*\) without any symmetry assumptions on the initial data and show that there is a family of perturbations which lead to blowup via \(u^*\). In similarity coordinates, this family represents a codimension-1 Lipschitz manifold modulo translation symmetries. The stability analysis relies on delicate spectral analysis for a non-self-adjoint operator. In addition, in \(d=7\) and \(d=9\), we prove nonradial stability of the well-known ODE blow-up solution. Also, for the first time we establish persistence of regularity for the wave equation in similarity coordinates.Blowup for a damped wave equation with mass and general nonlinear memoryhttps://zbmath.org/1536.351002024-07-17T13:47:05.169476Z"Feng, Zhendong"https://zbmath.org/authors/?q=ai:feng.zhendong"Guo, Fei"https://zbmath.org/authors/?q=ai:guo.fei"Li, Yuequn"https://zbmath.org/authors/?q=ai:li.yuequnSummary: We investigate the blowup conditions to the Cauchy problem for a semilinear wave equation with scale-invariant damping, mass and general nonlinear memory term (see Eq. (1.1) in the Introduction). We first establish a local (in time) existence result for this problem by Banach's fixed point theorem, where Palmieri's decay estimates on the solution to the corresponding linear homogeneous equation play an essential role in the proof. We then formulate a blowup result for energy solutions by applying the iteration argument together with the test function method.Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusionhttps://zbmath.org/1536.351032024-07-17T13:47:05.169476Z"Li, Yuequn"https://zbmath.org/authors/?q=ai:li.yuequn"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.5|liu.hui.2|liu.hui.1|liu.hui.4|liu.hui.6|liu.hui.9"Guo, Fei"https://zbmath.org/authors/?q=ai:guo.feiSummary: We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct.Blow-up and general decay of solutions for a Kirchhoff-type equation with distributed delay and variable-exponentshttps://zbmath.org/1536.351042024-07-17T13:47:05.169476Z"Ouchenane, Djamel"https://zbmath.org/authors/?q=ai:ouchenane.djamel"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Choucha, Abdelbaki"https://zbmath.org/authors/?q=ai:choucha.abdelbaki"Alnegga, Mohammad"https://zbmath.org/authors/?q=ai:alnegga.mohammadSummary: A nonlinear Kirchhoff-type equation with distributed delay and variableexponents is studied. Under suitable hypothesis the blow-up of solutions is proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case \(b = 0\).The blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative-typehttps://zbmath.org/1536.351052024-07-17T13:47:05.169476Z"Sasaki, Takiko"https://zbmath.org/authors/?q=ai:sasaki.takikoSummary: In this paper, we study a blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative type in one space dimension. Employing the idea of \textit{L. A. Caffarelli} and \textit{A. Friedman} [Trans. Am. Math. Soc. 297, 223--241 (1986; Zbl 0638.35053)], we prove the blow-up curve becomes Lipschitz continuous under suitable initial conditions. Moreover, we show the blow-up rates of the solution of the wave equations.Dispersion for the wave equation outside a cylinder in \(\mathbb{R}^3\)https://zbmath.org/1536.351112024-07-17T13:47:05.169476Z"Iandoli, Felice"https://zbmath.org/authors/?q=ai:iandoli.felice"Ivanovici, Oana"https://zbmath.org/authors/?q=ai:ivanovici.oanaSummary: We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in \(\mathbb{R}^3\) and we construct a global in time parametrix to derive sharp dispersion estimates for all frequencies (low and high) and, as a corollary, Strichartz estimates, all matching the \(\mathbb{R}^3\) case.A sharpened Strichartz inequality for the wave equationhttps://zbmath.org/1536.351122024-07-17T13:47:05.169476Z"Negro, Giuseppe"https://zbmath.org/authors/?q=ai:negro.giuseppeSummary: We disprove a conjecture of \textit{D. Foschi} [J. Eur. Math. Soc. (JEMS) 9, No. 4, 739--774 (2007; Zbl 1231.35028)], regarding extremizers for the Strichartz inequality with data in the Sobolev space \(\dot{H}^{1/2} \times \dot{H}^{-1/2} (\mathbb{R}^{d})\), for even \(d \ge 2\). On the other hand, we provide evidence to support the conjecture in odd dimensions and refine his sharp inequality in \(\mathbb{R}^{1 + 3}\), adding a term proportional to the distance of the initial data from the set of extremizers. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform.\(T_5\) configurations and hyperbolic systemshttps://zbmath.org/1536.351362024-07-17T13:47:05.169476Z"Johansson, Carl Johan Peter"https://zbmath.org/authors/?q=ai:johansson.carl-johan-peter"Tione, Riccardo"https://zbmath.org/authors/?q=ai:tione.riccardoSummary: In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in
[\textit{B. Kirchheim} et al., in: Geometric analysis and nonlinear partial differential equations. Berlin: Springer. 347--395 (2003; Zbl 1290.35097), Section 7],
and many of its properties have already been shown in
[\textit{A. Lorent} and \textit{G. Peng}, Arch. Ration. Mech. Anal. 234, No. 2, 857--910 (2019; Zbl 1479.49021); Calc. Var. Partial Differ. Equ. 59, No. 5, Paper No. 156, 36 p. (2020; Zbl 1448.35333)].
In particular, in
[Lorent and Peng, 2020, loc. cit.],
it is shown that the differential inclusion does not contain any \(T_4\) configurations. Here, we continue that study by showing that the differential inclusion does not contain \(T_5\) configurations.Blow-up and energy decay for a class of wave equations with nonlocal Kirchhoff-type diffusion and weak dampinghttps://zbmath.org/1536.351972024-07-17T13:47:05.169476Z"Liao, Menglan"https://zbmath.org/authors/?q=ai:liao.menglan"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong.1|tan.zhongSummary: The purpose of this paper is to study the following equation driven by a nonlocal integro-differential operator \(\mathcal{L}_K\):
\[
u_{tt} + [u]_s^{2(\theta -1)}\mathcal{L}_Ku + a|u_t|^{m-1}u_t = b|u|^{p-1}u,
\]
with homogeneous Dirichlet boundary condition and initial data, where \([u]_s^2\) is the Gagliardo seminorm, \(a \geq 0\), \(b > 0\), \(0 < s < 1\), and \(\theta\in[1, 2_s^\ast/2)\) with \(2_s^\ast = 2N/(N - 2s)\), \(N\) is the space dimension. By virtue of a differential inequality technique, an upper bound of the blow-up time is obtained with a bounded initial energy if \(m < p\) and some additional conditions are satisfied. For \(m\equiv 1\), in particular, the blow-up result with high initial energy also is showed by constructing a new control functional and combining energy inequalities with the concavity argument. Moreover, an estimate for the lower bound of the blow-up time is investigated. Finally, the energy decay estimate is proved as well. These results improve and complement some recent works.
{\copyright} 2023 John Wiley \& Sons, Ltd.Explicit formula of radiation fields of free waves with applications on channel of energyhttps://zbmath.org/1536.351982024-07-17T13:47:05.169476Z"Li, Liang"https://zbmath.org/authors/?q=ai:li.liang|li.liang.4"Shen, Ruipeng"https://zbmath.org/authors/?q=ai:shen.ruipeng"Wei, Lijuan"https://zbmath.org/authors/?q=ai:wei.lijuanSummary: We give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel-of-energy theory. We characterize all the radial weakly nonradiative solutions in all dimensions and give a few new exterior energy estimates.An \(r^p\)-weighted local energy approach to global existence for null form semilinear wave equationshttps://zbmath.org/1536.351992024-07-17T13:47:05.169476Z"Facci, Michael"https://zbmath.org/authors/?q=ai:facci.michael"McEntarrfer, Alex"https://zbmath.org/authors/?q=ai:mcentarrfer.alex"Metcalfe, Jason"https://zbmath.org/authors/?q=ai:metcalfe.jason-lSummary: We revisit the proof of small-data global existence for semilinear wave equations that satisfy a null condition. This new approach relies on a weighted local energy estimate that is akin to those of Dafermos and Rodnianski. Using weighted Sobolev estimates to obtain spatial decay and arguing in the spirit of the work of Keel, Smith, and Sogge, we are able to obtain global existence while only relying on translational and (spatial) rotational symmetries.Solvability results for the transient acoustic scattering by an elastic obstaclehttps://zbmath.org/1536.352002024-07-17T13:47:05.169476Z"Bonnet, Marc"https://zbmath.org/authors/?q=ai:bonnet.marc"Chaillat, Stéphanie"https://zbmath.org/authors/?q=ai:chaillat.stephanie"Nassor, Alice"https://zbmath.org/authors/?q=ai:nassor.aliceSummary: The well-posedness of the linear evolution problem governing the transient scattering of acoustic waves by an elastic obstacle is investigated. After using linear superposition in the acoustic domain, the analysis focuses on an equivalent causal transmission problem. The proposed analysis provides existence and uniqueness results, as well as continuous data-to-solution maps. Solvability results are established for three cases, which differ by the assumed regularity in space on the transmission data on the acoustic-elastic interface \(\Gamma\). The first two results consider data with ``standard'' \(H^{- 1/2}(\Gamma)\) and improved \(H^{1/2}(\Gamma)\) regularity in space, respectively, and are established using the Hille-Yosida theorem and energy identities. The third result assumes data with \(L^2(\Gamma)\) regularity in space and follows by Sobolev interpolation. Obtaining the latter result was motivated by the key role it plays (in a separate study) in the justification of an iterative numerical solution method based on domain decomposition. A numerical example is presented to emphasize the latter point.A mixed problem for a class of second-order nonlinear hyperbolic systems with Dirichlet and Poincaré boundary conditionshttps://zbmath.org/1536.352012024-07-17T13:47:05.169476Z"Dzhokhadze, O. M."https://zbmath.org/authors/?q=ai:dzhokhadze.otar-mikhajlovich"Kharibegashvili, S. S."https://zbmath.org/authors/?q=ai:kharibegashvili.s-s"Shavlakadze, N. N."https://zbmath.org/authors/?q=ai:shavlakadze.n-nSummary: For a certain class of second-order hyperbolic systems, a mixed problem with Dirichlet and Poincaré boundary conditions is studied. In the linear case, an explicit representation of a soultion of this problem is given and questions related to its uniqueness and existence are studied depending on the character of nonlinearities in the system.A note on the exact boundary controllability for an imperfect transmission problemhttps://zbmath.org/1536.352022024-07-17T13:47:05.169476Z"Monsurrò, S."https://zbmath.org/authors/?q=ai:monsurro.sara"Nandakumaran, A. K."https://zbmath.org/authors/?q=ai:nandakumaran.akamabadath-k"Perugia, C."https://zbmath.org/authors/?q=ai:perugia.carmenSummary: In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.Nonlocal problems with integral conditions for hyperbolic equations with two time variableshttps://zbmath.org/1536.352032024-07-17T13:47:05.169476Z"Varlamova, Galina Aleksandrovna"https://zbmath.org/authors/?q=ai:varlamova.galina-aleksandrovna"Kozhanov, Aleksandr Ivanovich"https://zbmath.org/authors/?q=ai:kozhanov.aleksandr-ivanovichSummary: The work is devoted to the study of solvability of boundary value problems with nonlocal conditions of integral form for the differential equations
\[u_{xt} - a u_{xx} + c(x, t)u = f(x, t),\]
in which \(x \in \Omega = (0, 1)\), \(t \in (0, T)\), \(0 < T < + \infty \), \(a \in \mathbb{R}\), and \(c(x, t)\) and \(f(x, t)\) are known functions. The peculiarity of these equations is that any of variables \(t\) and \(x\) can be considered a temporary variable, and in accordance with this, for these equations, formulations of boundary value problems with different carriers of boundary conditions can be proposed. For the problems under study, the work proves existence and uniqueness theorems for regular solutions; namely, solutions that have all derivatives generalized according to S. L. Sobolev and included in the equation.Comparison of polynomials and weighted-hyperbolic operatorshttps://zbmath.org/1536.352042024-07-17T13:47:05.169476Z"Khachaturyan, M. A."https://zbmath.org/authors/?q=ai:khachaturyan.m-a"Margaryan, V. N."https://zbmath.org/authors/?q=ai:margaryan.vachagan-nSummary: In the language of zero multiplicities of subpolynomials, the sufficient conditions are found under which a polynomial of two variables is hyperbolic with a given weight when its leading part is Gårding hyperbolic.Exponential stabilization of a flexible structure: a delayed boundary force control versus a delayed boundary moment controlhttps://zbmath.org/1536.352052024-07-17T13:47:05.169476Z"Chentouf, Boumediène"https://zbmath.org/authors/?q=ai:chentouf.boumediene"Smaoui, Nejib"https://zbmath.org/authors/?q=ai:smaoui.nejibSummary: The main concern of this paper is to study the boundary stabilization problem of the disk-beam system. To do so, we assume that the boundary control is either of a force type control or a moment type control and is subject to the presence of a constant time-delay. First, we show that in both cases, the system is well-posed in an appropriate functional space. Next, the exponential stability property is established. Finally, the obtained outcomes are ascertained through numerical simulations.BV solutions to a hyperbolic system of balance laws with logistic growthhttps://zbmath.org/1536.352062024-07-17T13:47:05.169476Z"Chen, Geng"https://zbmath.org/authors/?q=ai:chen.geng|chen.geng.3"Zeng, Yanni"https://zbmath.org/authors/?q=ai:zeng.yanniSummary: We study BV solutions for a \(2 \times 2\) system of hyperbolic balance laws. We show that when initial data have small total variation on \((- \infty, \infty)\) and small amplitude, and decay sufficiently fast to a constant equilibrium state as \(| x | \to \infty\), a Cauchy problem (with generic data) has a unique admissible BV solution defined globally in time. Here the solution is admissible in the sense that its shock waves satisfy the Lax entropy condition. We also study asymptotic behavior of solutions. In particular, we obtain a time decay rate for the total variation of the solution, and a convergence rate of the solution to its time asymptotic solution. Our system is a modification of a Keller-Segel type chemotaxis model. Its flux function possesses new features when comparing to the well-known model of Euler equations with damping. This may help to shed light on how to extend the study to a general system of hyperbolic balance laws in the future.On the unique solvability of radiative transfer equations with polarizationhttps://zbmath.org/1536.352072024-07-17T13:47:05.169476Z"Bosboom, Vincent"https://zbmath.org/authors/?q=ai:bosboom.vincent"Schlottbom, Matthias"https://zbmath.org/authors/?q=ai:schlottbom.matthias"Schwenninger, Felix L."https://zbmath.org/authors/?q=ai:schwenninger.felix-lSummary: We investigate the well-posedness of the radiative transfer equation with polarization and varying refractive index. The well-posedness analysis includes non-homogeneous boundary value problems on bounded spatial domains, which requires the analysis of suitable trace spaces. Additionally, we discuss positivity, Hermiticity, and norm-preservation of the matrix-valued solution. As auxiliary results, we derive new trace inequalities for products of matrices.Weak asymptotic solution of one dimensional zero pressure dynamics system in the quarter planehttps://zbmath.org/1536.352082024-07-17T13:47:05.169476Z"Joseph, Kayyunnapara Divya"https://zbmath.org/authors/?q=ai:joseph.kayyunnapara-divyaSummary: In this paper we study a system of equations which appear in the modelling of many physical phenomena. Initially this system appeared in description of the large scale structure formation. Recently it is derived as a second order queueing model. We construct weakly asymptotic solutions of the initial boundary value problem for the system and interaction of waves in the quarter plane \(\lbrace (x,t): x>0,t>0\rbrace\) with boundary Riemann solution centered at (0,0) and Riemann solution centered at a point \((x_0,0)\), \(x_0>0\).
{\copyright} 2023 Wiley-VCH GmbH.A relaxation approach to modeling properties of hyperbolic-parabolic type modelshttps://zbmath.org/1536.352092024-07-17T13:47:05.169476Z"Abreu, Eduardo"https://zbmath.org/authors/?q=ai:abreu.eduardo"Santo, Arthur Espírito"https://zbmath.org/authors/?q=ai:espirito-santo.arthur"Lambert, Wanderson"https://zbmath.org/authors/?q=ai:lambert.wanderson-j"Pérez, John"https://zbmath.org/authors/?q=ai:perez.johnSummary: In this work, we propose a novel relaxation modeling approach for partial differential equations (PDEs) involving convective and diffusive terms. We reformulate the original convection-diffusion problem as a system of hyperbolic equations coupled with relaxation terms. In contrast to existing literature on relaxation modeling, where the solution of the reformulated problem converges to certain types of equations in the diffusive limit, our formalism treats the augmented problem as a system of coupled hyperbolic equations with relaxation acting on both the convective flux and the source term. Furthermore, we demonstrate that the new system of equations satisfies Liu's sub-characteristic condition. To verify the robustness of our proposed approach, we perform numerical experiments on various important models, including nonlinear convection-diffusion problems with discontinuous coefficients. The results show the promising potential of our relaxation modeling approach for both pure and applied mathematical sciences, with applications in different models and areas.Two-phase states of the generalized van der Waals gas: conditions of absolute and neutral shock wave stabilityhttps://zbmath.org/1536.352102024-07-17T13:47:05.169476Z"Blokhin, Alexander M."https://zbmath.org/authors/?q=ai:blokhin.aleksandr-mikhailovich"Rudometova, Anna S."https://zbmath.org/authors/?q=ai:rudometova.anna-s"Tkachev, Dmitry L."https://zbmath.org/authors/?q=ai:tkachev.dmitrii-leonidovichSummary: In this work we discuss an approach to the construction of the areas of two-phase states for real gases, state equations for which are from the article by Fogelson and Likhachev in 2004 and formulate conditions that define absolute and neutral stability of the shock waves.
{\copyright} 2022 Wiley-VCH GmbH.Delta shock wave as limits of vanishing viscosity for zero-pressure gas dynamics with energy conservation lawhttps://zbmath.org/1536.352112024-07-17T13:47:05.169476Z"Li, Shiwei"https://zbmath.org/authors/?q=ai:li.shiweiSummary: This paper studies the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. By the vanishing viscosity method, the stability of the solutions involving delta shock wave with Dirac delta functions developing in both state variables is established.
{\copyright} 2022 Wiley-VCH GmbH.Delta-shocks as limits of vanishing viscosity for a nonhomogeneous hyperbolic systemhttps://zbmath.org/1536.352122024-07-17T13:47:05.169476Z"Li, Shiwei"https://zbmath.org/authors/?q=ai:li.shiweiSummary: The Riemann problem for a \(2\times 2\) hyperbolic system of conservation laws with time-dependent damping is discussed. Making use of the variable substitution method, we construct six kinds of Riemann solutions including rarefaction wave, contact discontinuity, shock wave as well as delta-shock. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of delta-shock solution are established. Because of the appearing of the damping, the Riemann solutions are no longer self-similar. All wave curves are monotonic and have convexity-concavity. Moreover, by employing the vanishing viscosity method, we introduce a time-dependent viscous system to prove the stability of the solutions containing the delta-shocks.Exact solution for Riemann problems of the shear shallow water modelhttps://zbmath.org/1536.352132024-07-17T13:47:05.169476Z"Nkonga, Boniface"https://zbmath.org/authors/?q=ai:nkonga.boniface"Chandrashekar, Praveen"https://zbmath.org/authors/?q=ai:chandrashekar.praveenSummary: The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.Interaction of an acceleration wave with a characteristic shock in interstellar gas cloudshttps://zbmath.org/1536.352142024-07-17T13:47:05.169476Z"Sharma, Kajal"https://zbmath.org/authors/?q=ai:sharma.kajal"Arora, Rajan"https://zbmath.org/authors/?q=ai:arora.rajanSummary: In this paper, we have considered the van der Waals equation of state to study the one-dimensional spherically symmetric self-gravitating interstellar gas clouds. The transport equation for the acceleration waves is obtained. The evolution and propagation of the characteristic shock are considered to study its interaction with the acceleration wave. The amplitudes of the reflected and the transmitted waves are also determined. The effect of non-ideal parameter on the amplitude of the acceleration waves is discussed in detail.Stabilization of a semilinear wave equation with delayhttps://zbmath.org/1536.352152024-07-17T13:47:05.169476Z"Gonzalez Martinez, Victor Hugo"https://zbmath.org/authors/?q=ai:gonzalez-martinez.victor-hugo"Marchiori, Talita Druziani"https://zbmath.org/authors/?q=ai:marchiori.talita-druziani"de Souza Franco, Alisson Younio"https://zbmath.org/authors/?q=ai:de-souza-franco.alisson-younioSummary: We study the wellposedness and the stabilization of solutions of a semilinear wave equation with delay and locally distributed dissipation. The novelty of this paper is that we deal with the semilinear wave equation subject to delay and locally distributed damping without smallness conditions in the initial data or in the delay term. In order to address this, the argumentation requires the use of Strichartz estimates and some microlocal analysis results such as propagation of microlocal defect measures and the Gárard's linearizability property. To obtain the observability estimate in the critical case we prove a Unique Continuation Property for the semilinear wave equation and apply it to our problem. Once we establish essential observability properties for the solutions, it is not difficult to prove that the solutions decay exponentially to 0.Some exact and approximate solutions to a generalized Maxwell-Cattaneo equationhttps://zbmath.org/1536.352162024-07-17T13:47:05.169476Z"Herron, Isom H."https://zbmath.org/authors/?q=ai:herron.isom-h-jun"Mickens, Ronald E."https://zbmath.org/authors/?q=ai:mickens.ronald-eSummary: The simple heat conduction equation in one-space dimension does not have the property of a finite speed for information transfer. A partial resolution of this difficulty can be obtained within the context of heat conduction by the introduction of a partial differential equation (PDE) called the Maxwell-Cattaneo (M-C) equation, elsewhere called the damped wave equation, a special case of the telegraph equation. We construct a generalization to the M-C equation by allowing the relaxation time parameter to be a function of temperature. In the balance of the paper, we present a variety of special exact and approximate solutions to this nonlinear PDE.
{\copyright} 2023 Wiley Periodicals LLC.On the study of hyperbolic \(p(.)\)-bi-Laplace equation with variable exponenthttps://zbmath.org/1536.352172024-07-17T13:47:05.169476Z"Chaoui, Abderrazak"https://zbmath.org/authors/?q=ai:chaoui.abderrazakSummary: A high-order hyperbolic \(p(.)\)-bi-Laplace equation with variable exponent is studied. The well-posedness at each time step of the problem in suitable Lebesgue Sobolev spaces with variable exponent with the help of nonlinear monotone operators theory is investigated. Some a priori estimates are proved, from which we extract convergence and existence results using Galerkin-finite element method.Solutions of Weinstein type equations, Carleson measures and \(\mathrm{BMO}(\mathbb{R}_+, dm_\lambda)\)https://zbmath.org/1536.352182024-07-17T13:47:05.169476Z"Guo, Qingdong"https://zbmath.org/authors/?q=ai:guo.qingdong"Betancor, Jorge J."https://zbmath.org/authors/?q=ai:betancor.jorge-j"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyongSummary: Let \(\lambda \in (0, \infty)\). Consider the following Weinstein type equation
\[
\mathbb{L}_\lambda u(x, t) : = \partial_t^2 u(x, t) - \Delta_\lambda u(x, t) = 0, \quad (x, t) \in (0, \infty) \times (0, \infty),
\]
where \(\Delta_\lambda : = - \frac{d^2}{dx^2} - \frac{2 \lambda}{x} \frac{d}{dx}\) is the Bessel operator on \(\mathbb{R}_+ := (0, \infty)\). In this paper, the authors first establish an \(m_\lambda\)-Carleson characterization of \(\mathrm{BMO}(\mathbb{R}_+, d m_\lambda)\) via the Poisson semigroup associated with \(\Delta_\lambda\), where \(dm_\lambda(x) : = x^{2 \lambda} dx\). Based on this result, the authors further show that a function \(u\) is the Poisson integral of a function \(f \in \mathrm{BMO}(\mathbb{R}_+, d m_\lambda)\) if and only if \(u\) satisfies the Weinstein type equation and the \(m_\lambda\)-Carleson type condition
\[
\sup_{I \subset \mathbb{R}_+} \frac{1}{m_\lambda (I)} \int\limits_0^{| I |} \int\limits_I t |\nabla_{x, t} u(x, t) |^2 dm_\lambda(x) dt < \infty, \quad \nabla_{x, t} := (\partial_x, \partial_t).
\]Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulencehttps://zbmath.org/1536.352612024-07-17T13:47:05.169476Z"Colombo, Maria"https://zbmath.org/authors/?q=ai:colombo.maria"Crippa, Gianluca"https://zbmath.org/authors/?q=ai:crippa.gianluca"Sorella, Massimo"https://zbmath.org/authors/?q=ai:sorella.massimoThis paper studies anomalous dissipation and non-uniqueness phenomena in linear transport theory, for the equation
\[
\partial_t \theta + u \cdot \nabla \theta = 0 \quad \text{ in } \quad ]0,T[ \times \mathbb{T}^2 \, , \tag{1}
\]
posed for a scalar variable \(\theta:]0,T[\times \mathbb{T}^2\rightarrow \mathbb{R}\) and the velocity field \(u:]0,T[\times \mathbb{T}^2\rightarrow \mathbb{R}^2\), where \(\mathbb{T}^2 \) denotes the two-dimensional torus and \((0,T)\) is the time-interval. Periodic spacial extension is assumed and \(\theta\) is additionally subject to the initial condition \(\theta = \theta_{\mathrm{in}}\) at initial time. The velocity field is assumed to be Hölderian in space and, according to the context, either \(p\)-integrable or Hölderian in time, this means \(u \in L^p(]0,T[ ; C^{\alpha}(\mathbb{T}^2))\) or \(u \in C^{\alpha}(]0,T[\times \mathbb{T}^2)\). Moreover, it ought to be divergence-free in the weak sense. An important concern of the paper is to study in connection with (1) the asymptotic behaviour of the vanishing diffusivity solutions \(\theta = \theta_{\kappa}\) to the parabolic equation
\[
\partial_t \theta + u \cdot \nabla \theta = \kappa \, \Delta \theta \quad \text{ in } \quad ]0,T[ \times \mathbb{T}^2 \, , \tag{2}
\]
subject to periodic boundary conditions and the initial condition \(\theta_{\mathrm{in}}\).
In a first Theorem A, the authors show that for \(p \geq 2\) and \(0 \leq \alpha <1\), there exist a velocity field \(u \in L^p(]0,T[ ; C^{\alpha}(\mathbb{T}^2))\) and an initial value \(\theta_{\mathrm{in}} \in C^{\infty}(\mathbb{T}^2)\) such that with \(p^{\circ} = 2p/(p-1)\) and any \(0 \leq \beta < 1/2\) satisfying \(\alpha + 2\beta < 1\), the problem (1) admits a solution \(\theta \in L^{p^{\circ}}(0,T; \, C^{\beta}(\mathbb{T}^2))\) that strictly dissipates the \(L^2\)-norm, in the sense that \(\|\theta(T,\cdot)\|_{L^2(\mathbb{T}^2)} < \|\theta_{\mathrm{in}}\|_{L^2(\mathbb{T}^2)}\). This solution can be gained as the limit of the sequence \(\{\theta_{\kappa}\}\) of solutions to (2) that exhibit the phenomenon anomalous dissipation:
\[
\limsup_{\kappa \rightarrow 0} \kappa \, \int_0^T\int_{\mathbb{T}^2} |\nabla \theta_{\kappa}|^2 \, dxdt > 0 \, .
\]
In a second surprising Theorem B, the authors moreover show that for every \(\alpha \in [0,1[\) there are a velocity field \(u \in C^{\alpha}(]0,T[ \times \mathbb{T}^2)\) and initial data \(\theta_{\mathrm{in}} \in C^{\infty}(\mathbb{T}^2)\) such that the problem (1) possesses at least two solutions, the one of which conserves the \(L^2\)-norm, while the other will not. Both solutions are accumulation points of the sequence \(\{\theta_{\kappa}\}\) of solutions to (2), the first without and the second exhibiting anomalous dissipation. In this sense, the vanishing diffusion limit would not provide in its own a valid selection criterium for (1).
The theorems rely on concrete, though highly technical, construction techniques for the singular velocity field and the initial data, while the convergence analysis is based on the introduction of an appropriate stochastic flow and the associated Feynman-Kac representation of the solution to (2).
Reviewer: Pierre-Étienne Druet (Darmstadt)Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory termhttps://zbmath.org/1536.353222024-07-17T13:47:05.169476Z"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yuming"Wang, Hongli"https://zbmath.org/authors/?q=ai:wang.hongli"Yang, Bin"https://zbmath.org/authors/?q=ai:yang.bin.3Summary: This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor.Global solutions and relaxation limit to the Cauchy problem of a hydrodynamic model for semiconductorshttps://zbmath.org/1536.353252024-07-17T13:47:05.169476Z"Lu, Yun-guang"https://zbmath.org/authors/?q=ai:lu.yunguangSummary: In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field \(E\) and the a-priori \(L^\infty\) estimate, \(0 < 2 \delta \leq \rho^{\varepsilon , \delta} \leq M(t)\), \(| u^{\varepsilon , \delta} | \leq M(t)\), where \(M(t)\) could tend to infinity as the time \(t\) tends to infinity, on the viscosity-flux approximate solutions \(( \rho^{\varepsilon , \delta}, u^{\varepsilon , \delta})\); Second, the compensated compactness theory is applied to prove the pointwise convergence of \(( \rho^{\varepsilon , \delta}, u^{\varepsilon , \delta})\) as \(\varepsilon, \delta\) go to zero, and that the limit \((\rho(x, t), u(x, t))\) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and \(\int_{- \infty}^x \rho^{\varepsilon , \delta}(x, t) - 2 \delta d x\), deduces the uniform \(L^\infty\) estimate, \(0 < 2 \delta \leq \rho^{\varepsilon , \delta} \leq M\), \(| u^{\varepsilon , \delta} | \leq M\), independent of the time \(t\) and \(\varepsilon, \delta \); Finally, as a by-product, the known compactness framework
[\textit{S. Junca} and \textit{M. Rascle}, Q. Appl. Math. 58, No. 3, 511--521 (2000; Zbl 1127.35354); \textit{P. Marcati} and \textit{R. Natalini}, Arch. Ration. Mech. Anal. 129, No. 2, 129--145 (1995; Zbl 0829.35128)]
is applied to show the relaxation limit, as the relation time \(\tau\) and \(\varepsilon, \delta\) go to zero, for general pressure \(P(\rho)\).Nonlinear acoustic equations of fractional higher order at the singular limithttps://zbmath.org/1536.353432024-07-17T13:47:05.169476Z"Nikolić, Vanja"https://zbmath.org/authors/?q=ai:nikolic.vanjaSummary: When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear acoustic equations of fractional higher order. In this work, we relate them to the classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxation times. To this end, we perform a singular limit analysis and determine their behavior as the relaxation time tends to zero. We show that, depending on the nonlinearities and assumptions on the data, these models can be seen as approximations of the Westervelt, Blackstock, or Kuznetsov wave equations in nonlinear acoustics. We furthermore establish the convergence rates and thus determine the error one makes when exchanging local and nonlocal models. The analysis rests upon the uniform bounds for the solutions of the acoustic equations with fractional higher-order derivatives, obtained through a testing procedure tailored to the coercivity property of the involved (weakly) singular memory kernel.Approximate solutions to hyperbolic partial differential equation with fractional differential and fractional integral forcing functionshttps://zbmath.org/1536.353552024-07-17T13:47:05.169476Z"Gupta, Nishi"https://zbmath.org/authors/?q=ai:gupta.nishi"Maqbul, Md."https://zbmath.org/authors/?q=ai:maqbul.mdSummary: This manuscript deals with a hyperbolic partial differential equation with fractional differential and fractional integral forcing functions. Semidiscretization method is used to establish a unique strong solution and also approximate solutions. Error estimates and continuous dependence of the strong solution on the given conditions have also been discussed. At the end, we illustrated the results with an example.Tangential cone condition for the full waveform forward operator in the viscoelastic regime: the nonlocal casehttps://zbmath.org/1536.353712024-07-17T13:47:05.169476Z"Eller, Matthias"https://zbmath.org/authors/?q=ai:eller.matthias-m"Griesmaier, Roland"https://zbmath.org/authors/?q=ai:griesmaier.roland"Rieder, Andreas"https://zbmath.org/authors/?q=ai:rieder.andreasSummary: We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of \textit{M. Eller} and \textit{A. Rieder} [Inverse Probl. 37, No. 8, Article ID 085011, 17 p. (2021; Zbl 1486.65149)] from the acoustic to the viscoelastic wave equation. In particular, we establish injectivity of the Fréchet derivative of the parameter-to-state map for a semidiscrete seismic inverse problem in the viscoelastic regime. Here the finite-dimensional parameter space is restricted to functions having global support in the propagation medium (the nonlocal case) and that are locally linearly independent. As a consequence, we deduce local conditional well-posedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear ill-posed problems.Solvability of problems of recovering the external influence in the first order hyperbolic equationshttps://zbmath.org/1536.353752024-07-17T13:47:05.169476Z"Kozhanov, Aleksandr Ivanovich"https://zbmath.org/authors/?q=ai:kozhanov.aleksandr-ivanovich"Aĭtzhanov, Serik Ersultanovich"https://zbmath.org/authors/?q=ai:aitzhanov.serik-ersultanovich"Zhalgasova, Korkem Abildakyzy"https://zbmath.org/authors/?q=ai:zhalgasova.korkem-abildakyzySummary: We study the solvability in Sobolev spaces of the problem of recovering the coefficients of the right-hand side, or the external influence, in the first order hyperbolic differential equations. Such problems belong to the class of linear inverse problems for partial differential equations. For the problems under study, we prove the existence and uniqueness theorems for regular solutions (having all generalized in Sobolev's sense derivatives entering the equation).Inverse problems of restoring parameters in parabolic and hyperbolic equationshttps://zbmath.org/1536.353762024-07-17T13:47:05.169476Z"Kozhanov, Aleksandr Ivanovich"https://zbmath.org/authors/?q=ai:kozhanov.aleksandr-ivanovich"Telesheva, Lyubov' Aleksandrovna"https://zbmath.org/authors/?q=ai:telesheva.lyubov-aleksandrovnaSummary: The work is devoted to the study of the solvability of new inverse problems of determining, together with the solution of parabolic or hyperbolic equations, a certain coefficient of the equation itself. A feature of the problems under study is, firstly, that the unknown coefficient is sought in the class of constant functions and, secondly, that a new, previously unused redefinition condition is applied. For the problems under study, existence theorems are proved for regular solutions, which are the solutions having all the derivatives generalized in the Sobolev sense entering the corresponding equation.A source identification problem in a bi-parabolic equation: convergence rates and some optimal resultshttps://zbmath.org/1536.353792024-07-17T13:47:05.169476Z"Mondal, Subhankar"https://zbmath.org/authors/?q=ai:mondal.subhankar"Nair, M. Thamban"https://zbmath.org/authors/?q=ai:nair.m-thambanSummary: This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the Hölder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.Maximum-principle-preserving, steady-state-preserving and large time-stepping high-order schemes for scalar hyperbolic equations with source termshttps://zbmath.org/1536.650842024-07-17T13:47:05.169476Z"Liu, Lele"https://zbmath.org/authors/?q=ai:liu.lele"Zhang, Hong"https://zbmath.org/authors/?q=ai:zhang.hong.6"Qian, Xu"https://zbmath.org/authors/?q=ai:qian.xu"Song, Songhe"https://zbmath.org/authors/?q=ai:song.songheSummary: In this paper, we construct a family of temporal high-order parametric relaxation Runge-Kutta (pRRK) schemes for stiff ordinary differential equations (ODEs), and explore their application in solving hyperbolic conservation laws with source terms. The new time discretization methods are explicit, large time-stepping, delay-free and able to preserve steady state. They are combined with fifth-order weighted compact nonlinear schemes (WCNS5) spatial discretization and parametrized maximum-principle-preserving (MPP) flux limiters to solve scalar hyperbolic equations with source terms. We prove that the fully discrete schemes preserve the maximum principle strictly. Through benchmark test problems, we demonstrate that the proposed schemes have fifth-order accuracy in space, fourth-order accuracy in time and allow for large time-stepping without time delay. Both theoretical analyses and numerical experiments are presented to validate the benefits of the proposed schemes.A shock-stable numerical scheme accurate for contact discontinuities: applications to 3D compressible flowshttps://zbmath.org/1536.650942024-07-17T13:47:05.169476Z"Hu, Lijun"https://zbmath.org/authors/?q=ai:hu.lijun"Wang, Xiaohui"https://zbmath.org/authors/?q=ai:wang.xiaohuiThe authors study the mechanism for three-dimensional shock instability by means of linearized stability analysis and the dissipation-controlling approach which introduces the required transverse dissipation in the numerical shock layer and is adopted to enhance the Roe scheme's shock stability. The corresponding innovative healing method based on a combination of a novel Roe-type flux and the second-order MUSCL reconstruction is proposed. The deficiency of unphysical expansion shocks is remedied by incorporating the adjacent cell's information in the calculation of numerical signal wave speeds. In order to improve the accuracy for contact discontinuities and shear waves, the authors use an algebraic method combining the tangent of hyperbola for interface capturing scheme and the boundary variation diminishing algorithm to further minimize the density difference in the numerical diffusion term. A series of numerical experiments fully demonstrates an excellent robustness against shock instability and high accuracy for contact discontinuities and shear waves.
Reviewer: Ljiljana Teofanov (Novi Sad)DynAMO: multi-agent reinforcement learning for dynamic anticipatory mesh optimization with applications to hyperbolic conservation lawshttps://zbmath.org/1536.650982024-07-17T13:47:05.169476Z"Dzanic, T."https://zbmath.org/authors/?q=ai:dzanic.tarik"Mittal, K."https://zbmath.org/authors/?q=ai:mittal.ketan"Kim, D."https://zbmath.org/authors/?q=ai:kim.dohyun.2"Yang, J."https://zbmath.org/authors/?q=ai:yang.jiachen"Petrides, S."https://zbmath.org/authors/?q=ai:petrides.socratis"Keith, B."https://zbmath.org/authors/?q=ai:keith.brendan"Anderson, R."https://zbmath.org/authors/?q=ai:anderson.robert-c|anderson.robert-j-jun|anderson.robert-f-v|anderson.robert-lee|anderson.robert-leonard|anderson.robert-d|anderson.robert-w-g|anderson.robert-b.1|anderson.robert-mSummary: We introduce DynAMO, a reinforcement learning paradigm for Dynamic Anticipatory Mesh Optimization. Adaptive mesh refinement is an effective tool for optimizing computational cost and solution accuracy in numerical methods for partial differential equations. However, traditional adaptive mesh refinement approaches for time-dependent problems typically rely only on instantaneous error indicators to guide adaptivity. As a result, standard strategies often require frequent remeshing to maintain accuracy. In the DynAMO approach, multi-agent reinforcement learning is used to discover new local refinement policies that can anticipate and respond to future solution states by producing meshes that deliver more accurate solutions for longer time intervals. By applying DynAMO to discontinuous Galerkin methods for the linear advection and compressible Euler equations in two dimensions, we demonstrate that this new mesh refinement paradigm can outperform conventional threshold-based strategies while also generalizing to different mesh sizes, remeshing and simulation times, and initial conditions.Finite element-based invariant-domain preserving approximation of hyperbolic systems: beyond second-order accuracy in spacehttps://zbmath.org/1536.651062024-07-17T13:47:05.169476Z"Guermond, Jean-Luc"https://zbmath.org/authors/?q=ai:guermond.jean-luc"Nazarov, Murtazo"https://zbmath.org/authors/?q=ai:nazarov.murtazo"Popov, Bojan"https://zbmath.org/authors/?q=ai:popov.boyanSummary: This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high-order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from [\textit{R. Abgrall} et al., J. Sci. Comput. 72, No. 3, 1232--1268 (2017; Zbl 1383.65138)]. The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting.Approximation of three-dimensional nonlinear wave equations by fundamental solutions and weighted residuals processhttps://zbmath.org/1536.651212024-07-17T13:47:05.169476Z"Safari, Farzaneh"https://zbmath.org/authors/?q=ai:safari.farzanehSummary: In this paper, the localized method of fundamental solutions (LMFS) with coupling the dual reciprocity method (DRM) is applied to simulate three-space dimensional nonlinear wave equations. First, DRM, which is a popular meshless method based on polyharmonic splines (PhS), is applied to obtain the particular solution. After evaluating the particular solution, 3D-LMFS method can be employed to evaluate the homogeneous solution. The proposed 3D-LMFS algorithms construct 3D artificial surfaces where source points and then the collocation procedure propose by using the fundamental solutions in each 3D surface. Straightforwardly, a solution to the 3D wave equation is approximated by a sum of the combination of PhS and fundamental solutions using LMFS. Eventually, the scheme has been prosperously tested with selected examples.
{\copyright} 2023 John Wiley \& Sons Ltd.On the Riemann problem and interaction of elementary waves for two-layered blood flow model through arterieshttps://zbmath.org/1536.761552024-07-17T13:47:05.169476Z"Jana, Sumita"https://zbmath.org/authors/?q=ai:jana.sumita"Kuila, Sahadeb"https://zbmath.org/authors/?q=ai:kuila.sahadebSummary: In this paper, we focus on the Riemann problem for two-layered blood flow model, which is represented by a system of quasi-linear hyperbolic partial differential equations (PDEs) derived from the Euler equations by vertical averaging across each layer. We consider the Riemann problem with varying velocities and equal constant density through arteries. For instance, the flow layer close to the wall of vessel has a slower average speed than the layer far from the vessel because of the viscous effect of the blood vessel. We first establish the existence and uniqueness of the corresponding Riemann solution by a thorough investigation of the properties of elementary waves, namely, shock wave, rarefaction wave, and contact discontinuity wave. Further, we extensively analyze the elementary wave interaction between rarefaction wave and shock wave with contact discontinuity and rarefaction wave and shock wave. The global structure of the Riemann solutions after each wave interaction is explicitly constructed and graphically illustrated towards the end.
{\copyright} 2023 John Wiley \& Sons Ltd.On hierarchical competition through reduction of individual growthhttps://zbmath.org/1536.920842024-07-17T13:47:05.169476Z"Barril, Carles"https://zbmath.org/authors/?q=ai:barril.carles"Calsina, Àngel"https://zbmath.org/authors/?q=ai:calsina.angel"Diekmann, Odo"https://zbmath.org/authors/?q=ai:diekmann.odo"Farkas, József Z."https://zbmath.org/authors/?q=ai:farkas.jozsef-zoltanSummary: We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.Exact internal controllability and exact internal synchronization for a kind of first order hyperbolic systemhttps://zbmath.org/1536.930822024-07-17T13:47:05.169476Z"Li, Tatsien"https://zbmath.org/authors/?q=ai:li.tatsien"Lu, Xing"https://zbmath.org/authors/?q=ai:lu.xing"Qu, Peng"https://zbmath.org/authors/?q=ai:qu.pengSummary: In this paper we investigate the exact controllability and exact synchronization in \(L^2\) space for a 1-D first order linear hyperbolic system with internal controls located on some part of the domain. Based on the exact boundary controllability theory, we first use the constructive method to establish the exact controllability by internal controls for a time reversible system with inhomogeneous boundary conditions. The method is then adapted to prove the exact internal null controllability for a general system with homogeneous boundary conditions. These results can be then used to establish the exact internal synchronization for the system, and related subjects are further studied.Observer design for a class of semilinear hyperbolic PDEs with distributed sensing and parametric uncertaintieshttps://zbmath.org/1536.932892024-07-17T13:47:05.169476Z"Holta, Haavard"https://zbmath.org/authors/?q=ai:holta.haavard"Aamo, Ole Morten"https://zbmath.org/authors/?q=ai:aamo.ole-mortenEditorial remark: No review copy delivered.Output regulation and tracking for linear ODE-hyperbolic PDE-ODE systemshttps://zbmath.org/1536.933762024-07-17T13:47:05.169476Z"Redaud, Jeanne"https://zbmath.org/authors/?q=ai:redaud.jeanne"Bribiesca-Argomedo, Federico"https://zbmath.org/authors/?q=ai:bribiesca-argomedo.federico"Auriol, Jean"https://zbmath.org/authors/?q=ai:auriol.jeanSummary: This paper proposes a constructive solution to the output regulation -- output tracking problem for a general class of interconnected systems. The class of systems under consideration consists of a linear \(2 \times 2\) hyperbolic partial differential equations (PDE) system coupled at both ends with ordinary differential equations (ODEs). The proximal ODE system, which represents actuator dynamics, is actuated. Colocated measurements are available. The distal ODE system represents the load dynamics. The control objective is to ensure, in the presence of a disturbance signal (regulation problem), that a virtual output exponentially converges to zero. By doing so, we can ensure that a state component of the distal ODE state robustly converges towards a known reference trajectory (output tracking problem) even in the presence of a disturbance with a known structure. The proposed approach combines the backstepping methodology and frequency analysis techniques. We first map the original system to a simpler target system using an invertible integral change of coordinates. From there, we design an adequate full-state feedback controller in the frequency domain. Following a similar approach, we propose a state observer that estimates the state and reconstructs the disturbance from the available measurement. Combining the full-state feedback controller with the state estimation results in a dynamic output-feedback control law. Finally, existing filtering techniques guarantee the closed-loop system robustness properties.Robust controllers for a flexible satellite modelhttps://zbmath.org/1536.933792024-07-17T13:47:05.169476Z"Govindaraj, Thavamani"https://zbmath.org/authors/?q=ai:govindaraj.thavamani"Humaloja, Jukka-Pekka"https://zbmath.org/authors/?q=ai:humaloja.jukka-pekka"Paunonen, Lassi"https://zbmath.org/authors/?q=ai:paunonen.lassiSummary: We consider a PDE-ODE model of a flexible satellite that is composed of two identical flexible solar panels and a center rigid body. We prove that the satellite model is exponentially stable in the sense that the energy of the solutions decays to zero exponentially. In addition, we construct two internal model based controllers, a passive controller and an observer based controller, such that the linear and angular velocities of the center rigid body converge to the given sinusoidal signals asymptotically. A numerical simulation is presented to compare the performances of the two controllers.Cooperative output regulation for networks of hyperbolic systems using adaptive cooperative observershttps://zbmath.org/1536.934192024-07-17T13:47:05.169476Z"Enderes, Tarik"https://zbmath.org/authors/?q=ai:enderes.tarik"Gabriel, Jakob"https://zbmath.org/authors/?q=ai:gabriel.jakob"Deutscher, Joachim"https://zbmath.org/authors/?q=ai:deutscher.joachimSummary: This paper is concerned with the cooperative output regulation problem for heterogeneous networks of hyperbolic agents. Both the leaders' output as well as the disturbances are generated by finite-dimensional signal models that are unknown to the followers. An adaptive cooperative reference observer is employed and an infinite-dimensional adaptive cooperative disturbance observer is designed using the backstepping approach to estimate both the states and the signal model parameters. By combining them with a local state feedback regulator, an adaptive cooperative output feedback regulator results. It is shown that the resulting network controlled multi-agent system achieves asymptotic leader-follower output synchronization. The results of the paper are illustrated for a heterogeneous network of three hyperbolic agents in simulations.Event-triggered output-feedback backstepping control of sandwich hyperbolic PDE systemshttps://zbmath.org/1536.935732024-07-17T13:47:05.169476Z"Wang, Ji"https://zbmath.org/authors/?q=ai:wang.ji.1"Krstic, Miroslav"https://zbmath.org/authors/?q=ai:krstic.miroslavEditorial remark: No review copy delivered.Singular perturbation analysis of a coupled system involving the wave equationhttps://zbmath.org/1536.936002024-07-17T13:47:05.169476Z"Cerpa, Eduardo"https://zbmath.org/authors/?q=ai:cerpa.eduardo"Prieur, Christophe"https://zbmath.org/authors/?q=ai:prieur.christopheEditorial remark: No review copy delivered.The active disturbance rejection control approach to output feedback stabilization of an anti-stable wave equation with corrupted boundary observationhttps://zbmath.org/1536.936652024-07-17T13:47:05.169476Z"Lang, Pei-Hua"https://zbmath.org/authors/?q=ai:lang.pei-hua"Feng, Hongyinping"https://zbmath.org/authors/?q=ai:feng.hongyinpingSummary: In this paper, we are concerned with the boundary output feedback stabilization for an anti-stable wave equation where the boundary observation is corrupted by a general disturbance. We cope with the disturbance by the approach of active disturbance rejection control. In contrast with the existing results, the assumptions on the derivative of the disturbance are removed. It is necessary to assume that the disturbance is Lipschitz continuous. A state observer together with a disturbance estimator is designed by using the corrupted output. Both the well-posedness and the stability of the closed-loop system are proved. The theoretical results are validated visually by some numerical simulations.Output feedback exponential stabilization for a one-dimensional wave equation with control matched nonlinear disturbancehttps://zbmath.org/1536.936692024-07-17T13:47:05.169476Z"Mei, Zhan-Dong"https://zbmath.org/authors/?q=ai:mei.zhandong"Zhou, Hua-Cheng"https://zbmath.org/authors/?q=ai:zhou.hua-chengEditorial remark: No review copy delivered.Limits of stabilization of a networked hyperbolic system with a circlehttps://zbmath.org/1536.937222024-07-17T13:47:05.169476Z"Gugat, Martin"https://zbmath.org/authors/?q=ai:gugat.martin"Huang, Xu"https://zbmath.org/authors/?q=ai:huang.xu"Wang, Zhiqiang"https://zbmath.org/authors/?q=ai:wang.zhi-qiangSummary: This paper is devoted to the discussion of the exponential stability of a networked hyperbolic system with a circle. Our analysis extends an example by \textit{G. Bastin} and \textit{J.-M. Coron} [Stability and boundary stabilization of 1-D hyperbolic systems. Basel: Birkhäuser/Springer (2016; Zbl 1377.35001)] about the limits of boundary stabilizability of hyperbolic systems to the case of a networked system that is defined on a graph which contains a cycle. By spectral analysis, we prove that the system is stabilizable while the length of the arcs is sufficiently small. However, if the length of the arcs is too large, the system is not stabilizable. Our results are robust with respect to small perturbations of the arc lengths. Complementing our analysis, we provide numerical simulations that illustrate our findings.ADRC dynamic stabilization of an unstable heat equationhttps://zbmath.org/1536.937372024-07-17T13:47:05.169476Z"Zhang, Yu-Long"https://zbmath.org/authors/?q=ai:zhang.yulong"Zhu, Min"https://zbmath.org/authors/?q=ai:zhu.min.4"Li, Donghai"https://zbmath.org/authors/?q=ai:li.donghai"Wang, Jun-Min"https://zbmath.org/authors/?q=ai:wang.junminEditorial remark: No review copy delivered.