Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses (Q1174279)

The asymptotic behaviour as \(t\to\infty\) of the solutions of the PDE \[ \partial u/\partial t-\Delta u=m(x,t,u)h(u) \hbox{ in }\Omega\times(0,\infty),\quad \partial u/\partial n=0 \hbox{ on } \partial\Omega\times(0,\infty) \] is studied under the following general assumptions: \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\); \(m\) is Hölder continuous in all its variables and periodic in \(t\) and \(h\) satisfies: \(h>0\) in \((a,b)\), \(h'(a)>0\), \(h'(b)<0\), \(h(a)=h(b)=0\). Sufficient conditions are formulated to obtain that every solution converges to some time periodic solution.