Asymptotic behaviour of solutions for porous medium equation with periodic absorption (Q5939815)

Consider the degenerate reaction-diffusion problem NEWLINE\[NEWLINE \begin{gathered} u_t=\Delta u^m +au^p \text{ in } \Omega\times(0,\infty), \\ u=0 \text{ on } \partial\Omega\times(0,\infty), \\ u(\cdot, 0)=u_0 \text{ in } \Omega \end{gathered} NEWLINE\]NEWLINE for constants \(m>1\) and \(p \geq 1\), \(a\) a smooth, positive, time-periodic function, \(\omega_0\) a smooth nonnegative function, and \(\Omega\) a bounded domain in \(\mathbb R^n\) with smooth boundary. The authors show that the behavior of the solution of this problem depends on the relation between \(m\) and \(p\) (which is well known when \(a\) is independent of time). In addition, when \(p=m\), the behavior depends also on the relation between \(a\) and the first eigenvalue \(\lambda_1\) of the Laplacian with zero Dirichlet data in \(\Omega\). Specifically, if \(p<m\), all solutions are asymptotically trapped between a maximal and a minimal nonnegative periodic solution of the differential equation and boundary condition. If \(p=m\) and \(a<\lambda_1\), then all solutions decay to zero like a (negative) power of \(t\); while if \(p=m\) and \(a>\lambda_1\), then there are choices of the initial data such that the solution blows up (possibly in infinite time). If \(p>m\) and \(p \leq m(n+2)/(n-2)\), then solutions with small initial data decay to zero like a power of \(t\). The proofs are fairly standard, using the maximum principle, regularity estimates, and test function arguments.