Time periodic solutions of porous medium equation

In this article, we study the time periodic solutions to the following porous medium equation under the homogeneous Dirichlet boundary condition:\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document} $$u_t-\Delta u^m = u^\alpha(a(x,t)-b(x,t)u^\beta)\quad {\rm{in}} \ \Omega\times (-\infty,+\infty).$$\end{document} The existence of nontrivial nonnegative solution is established provided that 0≤α< m . The existence is also proved in the case α= m but with an additional assumption \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document} $\mathop{\min}\nolimits_{\overline{\Omega}\times[0,T]}a(x,t){>}{\lambda}_1$\end{document} , where λ 1 is the first eigenvalue of the operator −Δ under the homogeneous Dirichlet boundary condition. We also show that the support of these solutions is independent of time by providing a priori estimates for their upper bounds using Moser iteration. Further, we establish the attractivity of maximal periodic solution using the monotonicity method. Copyright © 2010 John Wiley & Sons, Ltd.