Periodic parabolic equations on \(\mathbb{R}^ N\) and applications (Q1913881)

In contrast to the bounded domain very little is known about periodic-parabolic problems on unbounded domains. This paper concerns results on the existence of stable (with respect to the \(L_\infty\) norm) \(T\)-periodic solutions of the parabolic problem \[ \partial_t u- \Delta u= f(x, t, u)\quad \text{on } \mathbb{R}^N\times (0, \infty),\quad u(\cdot, 0)= u_0, \] where it is assumed that \(f: \mathbb{R}^N\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is sufficiently smooth, \(T\)-periodic in \(t\), and \(f(\cdot, \cdot, 0)\equiv 0\). The initial value \(u_0\) is taken from \(C_0(\mathbb{R}^N)\) or \(BUC(\mathbb{R}^N)\). The method of sub- and supersolution is used. Particular attention is devoted to Fischer's equation from population genetics in which \(f(x, t, u)= m(x, t)h(u)\). It is supposed that \(m\) is a smooth \(T\)-periodic function, which is negatively bounded away from zero at infinity, and \(h\in C^2(\mathbb{R})\) is a concave function, not necessarily strictly concave, satisfying \(h(0)= h(1)= 0\) and \(h'(0)> 0\). Since \(u\) is interpreted in population genetics as a relative density, only solutions with values in \([0, 1]\) are of interest. The main results are: (i) If the trivial solution \(u\equiv 0\) is linearly stable, then there is no nontrivial \(T\)-periodic solution and \(u\equiv 0\) is globally \(L_\infty\)-stable with respect to initial data in \(\{u_0\in BUC(\mathbb{R}^N): 0\leq u_0\ll 1\}\). (ii) If the trivial solution \(u\equiv 0\) is linearly unstable, then there exists a unique nontrivial \(T\)-periodic solution \(u^*\), which is globally asymptotically \(L_\infty\)-stable with respect to initial data in \(\{u_0\in BUC(\mathbb{R}^N): 0\leq u_0\ll 1\}\). (iii) Let the trivial solution \(u\equiv 0\) be neutrally stable. If \(h\) is not linear in some interval \([0, s_0]\) with \(s_0> 0\), then there is no nontrivial \(T\)-periodic solution, and \(u\equiv 0\) is globally asymptotically \(L_\infty\)-stable with respect to initial data in \(\{u_0\in BUC(\mathbb{R}^N): 0\leq u_0\ll 1\}\). If \(h\) is linear on such an interval \([0, s_0]\), there exists a one-parameter family \({\mathcal A}:= \{\varepsilon\phi: 0\leq \varepsilon\leq s_0\}\) of \(L_\infty\)-stable \(T\)-periodic solutions.