Recent zbMATH articles in MSC 47Dhttps://zbmath.org/atom/cc/47D2024-06-14T15:52:26.737412ZWerkzeugPeriodic solutions for third-order differential evolution equation set on singular domainhttps://zbmath.org/1534.340602024-06-14T15:52:26.737412Z"Chaouchi, Belkacem"https://zbmath.org/authors/?q=ai:chaouchi.belkacem"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this paper, we analyze the existence and uniqueness of periodic solutions for a third-order evolution differential equation set in singular cylindrical domain of \(\mathbb{R}^{3+1}\). The right-hand term of the equation is taken in some anisotropic Hölder spaces. Our strategy is based essentially on the study of a third order abstract differential equation. In order to achieve our aims, we essentially use the well known Dunford's operational calculus and some usual interpolation space.Strong solutions to quasi-linear differential equations with deviating argumentshttps://zbmath.org/1534.340712024-06-14T15:52:26.737412Z"Borah, Jayanta"https://zbmath.org/authors/?q=ai:borah.jayanta"Devi, Darshana"https://zbmath.org/authors/?q=ai:devi.darshana"Haloi, Rajib"https://zbmath.org/authors/?q=ai:haloi.rajibSummary: We use the Rothe's method to study the existence and uniqueness of a strong solution to a quasi-linear differential equation having deviating arguments in an arbitrary Banach space. Later, as an application, we provide an example to discuss the main result.A characterization of \(L^p\)-maximal regularity for time-fractional systems in \textit{UMD} spaces and applicationshttps://zbmath.org/1534.350492024-06-14T15:52:26.737412Z"Alvarez, Edgardo"https://zbmath.org/authors/?q=ai:alvarez.edgardo"Lizama, Carlos"https://zbmath.org/authors/?q=ai:lizama.carlosSummary: In this article we provide new insights into the well-posedness and maximal regularity of systems of abstract evolution equations, in the framework of periodic Lebesgue spaces of vector-valued functions. Our abstract model is flexible enough as to admit time-fractional derivatives in the sense of Liouville-Grünwald. We characterize the maximal regularity property solely in terms of \(R\)-boundedness of a block operator-valued symbol, and provide corresponding estimates. In addition, we show practical criteria that imply the \(R\)-boundedness part of the characterization. We apply these criteria to show that the Keller-Segel system, as well as a reactor model system, have \(L^q - L^p\) maximal regularity.Fourth-order Schrödinger type operator with unbounded coefficients in \(L^2(\mathbb{R}^N)\)https://zbmath.org/1534.351012024-06-14T15:52:26.737412Z"Gregorio, Federica"https://zbmath.org/authors/?q=ai:gregorio.federica"Tacelli, Cristian"https://zbmath.org/authors/?q=ai:tacelli.cristian(no abstract)Explicit improvements for \(\mathrm{L}^p\)-estimates related to elliptic systemshttps://zbmath.org/1534.351322024-06-14T15:52:26.737412Z"Böhnlein, Tim"https://zbmath.org/authors/?q=ai:bohnlein.tim"Egert, Moritz"https://zbmath.org/authors/?q=ai:egert.moritzSummary: We give a simple argument to obtain \(\mathrm{L}^p\)-boundedness for heat semigroups associated to uniformly strongly elliptic systems on \(\mathbb{R}^d\) by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give \(p\) explicitly in terms of ellipticity. It is optimal at the endpoint \(p=\infty\). We also obtain \(\mathrm{L}^p\)-estimates for the gradient of the semigroup, where \(p>2\) depends on ellipticity but not on dimension.
{\copyright} 2023 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.Remarks on the linear wave equationhttps://zbmath.org/1534.352582024-06-14T15:52:26.737412Z"Ball, John M."https://zbmath.org/authors/?q=ai:ball.john-mSummary: We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.Exponential stabilization of waves for the Zaremba boundary conditionhttps://zbmath.org/1534.352642024-06-14T15:52:26.737412Z"Cornilleau, Pierre"https://zbmath.org/authors/?q=ai:cornilleau.pierre-emmanuel"Robbiano, Luc"https://zbmath.org/authors/?q=ai:robbiano.lucSummary: We prove, under some geometrical condition on geodesic flow, exponential stabilization of wave equation with Zaremba boundary condition. We prove an estimate on the resolvent of semigroup associated with wave equation on the imaginary axis and we deduce the stabilization result. To prove this estimate we apply semiclassical measure techniques. The main difficulties are proving that support of measure is in characteristic set in a neighborhood of the jump in the boundary condition and proving results of propagation in a neighborhood of a boundary point where Neumann boundary condition is imposed. In fact a lot of results applied here are proved in previous articles; these two points are new.On the trace embedding and its applications to evolution equationshttps://zbmath.org/1534.460252024-06-14T15:52:26.737412Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Lindemulder, Nick"https://zbmath.org/authors/?q=ai:lindemulder.nick"Veraar, Mark"https://zbmath.org/authors/?q=ai:veraar.mark-cSummary: In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half-line are shown.
{{\copyright} 2023 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.}Bochner's subordination and fractional caloric smoothing in Besov and Triebel-Lizorkin spaceshttps://zbmath.org/1534.460362024-06-14T15:52:26.737412Z"Knopova, Victoriya"https://zbmath.org/authors/?q=ai:knopova.victoriya"Schilling, René L."https://zbmath.org/authors/?q=ai:schilling.rene-leanderSummary: We use Bochner's subordination technique to obtain caloric smoothing estimates in Besov- and Triebel-Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß-Weierstraß, Cauchy-Poisson and higher-order generalized Gauß-Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched.
{{\copyright} 2022 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH}Sharp solvability for singular SDEshttps://zbmath.org/1534.600732024-06-14T15:52:26.737412Z"Kinzebulatov, Damir"https://zbmath.org/authors/?q=ai:kinzebulatov.damir"Semënov, Yuliy A."https://zbmath.org/authors/?q=ai:semenov.yuliy-aSummary: The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a \(L^p\) variant of De Giorgi's method.Matrix-valued Schrödinger operators over finite adeleshttps://zbmath.org/1534.810482024-06-14T15:52:26.737412Z"Urban, R."https://zbmath.org/authors/?q=ai:urban.reena|urban.ryan|urban.romanSummary: Let \(K\) be an algebraic number field. With \(K\) we associate the ring of finite adeles \(\mathbb{A}_K\). In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group \(\mathbb{A}_K^n\). Specifically, we consider matrix-valued Hamiltonian operators \(H_{\mathbb{A}_K^n}= \Delta_{\mathbb{A}_K^n}\otimes\mathrm{Id}+V\), where \(\Delta_{\mathbb{A}_K^n}\) is the Vladimirov operator and \(V\) is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman-Kac formula for the Schrödinger semigroup generated by \(-H_{\mathbb{A}_K^n}\) This formula is given in terms of the ordered time exponentials.