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Zbl 0987.35086
Yin, Jingxue; Wang, Yifu
Asymptotic behaviour of solutions for nonlinear diffusion equation with periodic absorption. (English)
[A] Chen, Hua (ed.) et al., Partial differential equations and their applications. Proceedings of the conference, Wuhan, China, April 5-9, 1999. Singapore: World Scientific. 305-308 (1999). ISBN 981-02-4059-7/hbk

The article is devoted to studying a nonlinear diffusion equation with periodic absorption of the form $$\alignat 2 &\partial_tu=\Delta(u^m)+a(x,t)u^{\alpha}&&\quad\text{in }\Omega\times(0,\infty),\\ &u(x,t)= 0&&\quad\text{on }\partial\Omega\times(0,\infty),\\ &u(x,0)=u_0(x)&&\quad\text{in }\Omega,\endalignat$$ where $m>1$, $\alpha\geq 1$, $\Omega$ is a bounded domain in $\bbfR^N$ with smooth boundary, $a(x,t)$ is smooth, strictly positive and periodic in time with period $\omega>0$, and $u_0(x)$ is smooth and nonnegative. The aim of the article under review is to prove the existence of an attractor which consists of all nontrivial periodic solutions. In addition, the authors discuss the asymptotic behaviour of a multidimensional nonlinear diffusion equation.
[Vladimir N.Grebenev (Novosibirsk)]

MSC 2000:: *35K65 Parabolic equations of degenerate type
35B41 Attractors
34B40 Boundary value problems on infinite intervals
35K20 Second order parabolic equations, boundary value problems
35B10 Periodic solutions of PDE

Keywords: degenerate parabolic equation; attractor consisting of all nontrivial periodic solutions

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