Recent zbMATH articles in MSC 35K15https://zbmath.org/atom/cc/35K152021-06-15T18:09:00+00:00WerkzeugExistence of attractors for stochastic diffusion equations with fractional damping and time-varying delay.https://zbmath.org/1460.350462021-06-15T18:09:00+00:00"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xupingSummary: This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying delay. We then show pullback asymptotic compactness of solutions and the existence of random attractors by utilizing the Arzelà-Ascoli theorem and the uniform estimates for the derivative of the solution operator in the fractional Sobolev space \(H^\alpha(\mathbb{R}^n)\), with \(0 < \alpha < 1\).
{\copyright 2021 American Institute of Physics}
Reviewer: Reviewer (Berlin)Vanishing phenomena in fast decreasing generalized bistable equations.https://zbmath.org/1460.350382021-06-15T18:09:00+00:00"Li, Qi"https://zbmath.org/authors/?q=ai:li.qi|li.qi.1"Lou, Bendong"https://zbmath.org/authors/?q=ai:lou.bendongSummary: Consider the reaction diffusion equation \(u_t = u_{x x} + f(u)\) with generalized bistable nonlinearity: \(f(0) = f(\theta) = f(1) = 0\) for some \(\theta \in(0, 1)\), \(f(u) \leq 0\) in \((0, \theta)\), \(f(u) > 0\) in \((\theta, 1)\) and \(f(u) < 0\) in \((1, \infty)\). We show that when \(f(u)\) decreases sufficiently fast for \(u \gg 1\), there exists \(\varepsilon_0 > 0\) such that, for any nonnegative initial data \(u_0(x)\) with \(\mathrm{supp}( u_0) \subset [- \varepsilon_0, \varepsilon_0]\) (no matter how large \(\| u_0 \|_{L^\infty}\) is), the solution \(u(x, t)\) to the Cauchy problem with initial data \(u_0(x)\) always vanishes, that is, \(u(x, t) \to 0\) as \(t \to \infty\) in the \(L^\infty(\mathbb{R})\) norm.
Reviewer: Reviewer (Berlin)Inverse source problem for the abstract fractional differential equation.https://zbmath.org/1460.353932021-06-15T18:09:00+00:00"Kostin, Andrey B."https://zbmath.org/authors/?q=ai:kostin.andrey-b"Piskarev, Sergey I."https://zbmath.org/authors/?q=ai:piskarev.sergey-iSummary: In a Banach space, the inverse source problem for a fractional differential equation with Caputo-Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from \(D(A)\), and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator \(A\) is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.
Reviewer: Reviewer (Berlin)Asymptotic behavior of solutions to the logarithmic diffusion equation with a linear source.https://zbmath.org/1460.350412021-06-15T18:09:00+00:00"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahiko"Takáč, Peter"https://zbmath.org/authors/?q=ai:takac.peter"Yanagida, Eiji"https://zbmath.org/authors/?q=ai:yanagida.eijiSummary: We investigate the behavior of positive solutions to the Cauchy problem
\[
\begin{cases} \partial_t u =\partial_x^2(\log u)+u, & x\in\mathbb{R}, t>0, \\ \mathop{\lim}\limits_{x\rightarrow-\infty} \partial_x(\log u)=\alpha, \quad \mathop{\lim}\limits_{x\rightarrow+\infty}\partial_x(\log u)=\beta, \quad & t>0, \\ u(x,0)=u_0(x), & x\in\mathbb{R}, \end{cases}
\]
where \(\alpha,\beta \) are given positive constants and \(u_0(x)\) is a positive initial value. In the case of mass conservation, i.e., \(\int_{-\infty}^{\infty} u(x,t) dx \equiv \alpha + \beta \) for \(t\geq 0\), we show by an intersection number argument that the solution approaches a traveling wave as \(t\rightarrow \infty \). We then study the behavior in the case of mass expansion or extinction by using a transformation of variables. When the total mass is smaller, we show that extinction of the solution occurs in finite time and a rescaled solution converges to the traveling wave, whereas when the total mass is larger, the solution grows exponentially and a rescaled solution converges to a certain profile. Our results also include some log-concavity properties of solutions.
Reviewer: Reviewer (Berlin)On the well-posedness of a nonlinear pseudo-parabolic equation.https://zbmath.org/1460.352122021-06-15T18:09:00+00:00"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Au, Vo Van"https://zbmath.org/authors/?q=ai:au.vo-van"Tri, Vo Viet"https://zbmath.org/authors/?q=ai:tri.vo-viet"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper we consider the Cauchy problem for the pseudo-parabolic equation:
\[
\dfrac{\partial}{\partial t}(u+\mu(-\Delta)^{s_1}u)+(-\Delta)^{s_2}u=f(u),\quad x\in\Omega,\,t>0.
\]
Here, the orders \(s_1,s_2\) satisfy \(0<s_1\neq s_2 <1\) (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source \(f\) is globally Lipschitz. In the case when the source term \(f\) satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given.
Reviewer: Reviewer (Berlin)Finite-time blow-up for inhomogeneous parabolic equations with nonlinear memory.https://zbmath.org/1460.350512021-06-15T18:09:00+00:00"Alqahtani, Awatif"https://zbmath.org/authors/?q=ai:alqahtani.awatif"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: In this paper, we study the effect of an inhomogeneity \(w=w(x)\) on the finite-time blow-up of solutions to the nonlinear heat equation
\[
\partial_tu-\Delta u=\frac{1}{\Gamma(1-\gamma)}\int_0^t(t-s)^{-\gamma}|u(s)|^p\mathrm{ d}s+w(x),
\]
\[
(t,x)\in(0,T)\times\mathbb{R}^N,
\]
where \(N\geq 1\), \(0<\gamma<1\) and \(p>1\). It is well known that in the homogeneous case \(w\equiv 0\), the Fujita critical exponent is given by
\[
p_\ast=\max\left\{\frac{1}{\gamma}+\frac{4-2\gamma}{(N-2+2\gamma)^+}\right\}.
\]
In the case \(\int_{\mathbb{R}^N}w(x)dx>0\) and \(u(0,\cdot)\geq 0\), we prove that the critical exponent is equal to \(\infty\), which means that for all \(p>1\), we have a finite-time blow-up.
Reviewer: Reviewer (Berlin)High-order Wong-Zakai approximations for non-autonomous stochastic \(p\)-Laplacian equations on \(\mathbb{R}^N\).https://zbmath.org/1460.354042021-06-15T18:09:00+00:00"Zhao, Wenqiang"https://zbmath.org/authors/?q=ai:zhao.wenqiang"Zhang, Yijin"https://zbmath.org/authors/?q=ai:zhang.yijinSummary: In this paper, we investigate the approximations of stochastic \(p\)-Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random \(p\)-Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of \(q\)-times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.
Reviewer: Reviewer (Berlin)Large time behavior of ODE type solutions to parabolic \(p\)-Laplacian type equations.https://zbmath.org/1460.350352021-06-15T18:09:00+00:00"Eom, Junyong"https://zbmath.org/authors/?q=ai:eom.junyong"Sato, Ryuichi"https://zbmath.org/authors/?q=ai:sato.ryuichiSummary: Let \(u\) be a solution to the Cauchy problem for a nonlinear diffusion equation \[
\begin{cases}
\partial_t u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)+u^\alpha & \text{
in }\mathbb{R}^N\times(0, \infty), \\
u(x, 0)=\lambda+\varphi(x) & \text{ in }\mathbb{R}^N,
\end{cases}
\]
where \(N\geq 1\), \(2N/(N+1)<p\neq 2\), \(\alpha\in (-\infty, 1)\), \(\lambda>0\) and \(\varphi\in BC(\mathbb{R}^N)\cap L^1(\mathbb{R}^N)\) with \(\varphi\geq 0\) in \(\mathbb{R}^N\). Then the solution \(u\) behaves like a positive solution to ODE \(\zeta'=\zeta^\alpha\) in \((0,\infty)\). In this paper we show that the large time behavior of the solution \(u\) is described by a rescaled Barenblatt solution.
Reviewer: Reviewer (Berlin)Global solvability results for parabolic equations with \(p\)-Laplacian type diffusion.https://zbmath.org/1460.352062021-06-15T18:09:00+00:00"Chagas, J. Q."https://zbmath.org/authors/?q=ai:chagas.j-q"Guidolin, P. L."https://zbmath.org/authors/?q=ai:guidolin.p-l"Zingano, P. R."https://zbmath.org/authors/?q=ai:zingano.paulo-r-aSummary: We give conditions that assure global existence of bounded weak solutions to the Cauchy problem of general conservative 2nd-order parabolic equations with \(p\)-Laplacian type diffusion \((p>2)\) and initial data \(u_0\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\). Related results of interest are also given.
Reviewer: Reviewer (Berlin)Parabolic equations involving Laguerre operators and weighted mixed-norm estimates.https://zbmath.org/1460.350662021-06-15T18:09:00+00:00"Fan, Huiying"https://zbmath.org/authors/?q=ai:fan.huiying"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.taoSummary: In this paper, we study evolution equation \(\partial_tu=-L_\alpha u+f\) and the corresponding Cauchy problem, where \(L_\alpha\) represents the Laguerre operator \(L_\alpha=\frac{1}{2}(-\frac{d^2}{dx^2}+x^2+\frac{1}{x^2}(\alpha^2-\frac{1}{4}))\), for every \(\alpha\geq-\frac{1}{2}\). We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup \(\{e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0}\). In addition, we define the Poisson operator related to the fractional power \((\partial_t+L_\alpha)^s\) and reveal weighted mixed-norm estimates for revelent maximal operators.
Reviewer: Reviewer (Berlin)