Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Peter Poláčik

Displaying similar documents to “Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations”

Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Poláčik, Peter

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Asymptotic analysis for a nonlinear parabolic equation on $ℝ$

Eva Fašangová (1998)

Commentationes Mathematicae Universitatis Carolinae

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We show that nonnegative solutions of $\begin{matrix} u_{t} - u_{x x} + f (u) = 0, x \in ℝ, t > 0, \\ u = α \bar{u}, x \in ℝ, t = 0, supp \bar{u} compact \end{matrix}$ either converge to zero, blow up in $L^{2}$ -norm, or converge to the ground state when $t \to \infty$ , where the latter case is a threshold phenomenon when $α > 0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $α > 0$ , provided $supp \bar{u}$ is sufficiently small. ...

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

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In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the...

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

On the long-time behaviour of solutions of the p-Laplacian parabolic system

Paweł Goldstein (2008)

Colloquium Mathematicae

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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen, Tatiana Lobkova (2018)

Applications of Mathematics

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This paper is devoted to the study of the linear parabolic problem $ε \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x / ε, t / ε^{3}) \nabla u_{ε} (x, t)) = f (x, t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $ε$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic...

Multiscale homogenization of nonlinear hyperbolic-parabolic equations

Abdelhakim Dehamnia, Hamid Haddadou (2023)

Applications of Mathematics

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The main purpose of the present paper is to study the asymptotic behavior (when $ε \to 0$ ) of the solution related to a nonlinear hyperbolic-parabolic problem given in a periodically heterogeneous domain with multiple spatial scales and one temporal scale. Under certain assumptions on the problem’s coefficients and based on a priori estimates and compactness results, we establish homogenization results by using the multiscale convergence method.

Displaying similar documents to “Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations”