Asymptotic analysis for a nonlinear parabolic equation on $\mathbb {R}$

Eva Fašangová

Displaying similar documents to “Asymptotic analysis for a nonlinear parabolic equation on $ℝ$ ”

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

Peter Poláčik (2002)

Mathematica Bohemica

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We consider three types of semilinear second order PDEs on a cylindrical domain $Ω \times (0, \infty)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 2$ . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $Ω \times (0, \infty)$ is reserved for time $t$ , the third type is an elliptic equation with a singled out unbounded variable $t$ . We discuss the asymptotic behavior, as $t \to \infty$ , of solutions which are defined and bounded on $Ω \times (0, \infty)$ .

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.

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions $u$ : either the space derivative $u_{x}$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^{1}$ norm to the unique steady state. The main difficulty is to prove $C^{1}$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out...

$L^{\infty}$ estimates of solution for $m$ -Laplacian parabolic equation with a nonlocal term

Pulun Hou, Caisheng Chen (2011)

Czechoslovak Mathematical Journal

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In this paper, we consider the global existence, uniqueness and $L^{\infty}$ estimates of weak solutions to quasilinear parabolic equation of $m$ -Laplacian type $u_{t} - {div (| \nabla u |}^{m - 2} {\nabla u) = u | u |}^{β - 1} \int_{Ω} {| u |}^{α} d x$ in $Ω \times (0, \infty)$ with zero Dirichlet boundary condition in $\partial Ω$ . Further, we obtain the $L^{\infty}$ estimate of the solution $u (t)$ and $\nabla u (t)$ for $t > 0$ with the initial data $u_{0} \in L^{q} (Ω)$ $(q > 1)$ , and the case $α + β < m - 1$ .

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. ...

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Yūki Naito (2006)

Banach Center Publications

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We consider a nonlinear parabolic system modelling chemotaxis $u_{t} = \nabla \cdot (\nabla u - u \nabla v)$ , $v_{t} = Δ v + u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

Kentarou Fujie, Tomomi Yokota (2014)

Mathematica Bohemica

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This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $χ (v)$ and the growth term $f (u)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0 < χ (v) \leq χ_{0} / v^{k}$ $(k \geq 1$ , $χ_{0} > 0)$ and $λ_{1} - μ_{1} u \leq f (u) \leq λ_{2} - μ_{2} u$ $(λ_{1}, λ_{2}, μ_{1}, μ_{2} > 0)$ . It is shown that if $χ_{0}$ is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota. ...

Displaying similar documents to “Asymptotic analysis for a nonlinear parabolic equation on ℝ ”

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

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

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

L ∞ estimates of solution for m -Laplacian parabolic equation with a nonlocal term

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on ℝ N

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

Displaying similar documents to “Asymptotic analysis for a nonlinear parabolic equation on $ℝ$ ”

$L^{\infty}$ estimates of solution for $m$ -Laplacian parabolic equation with a nonlocal term

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