## Periodic parabolic equations on $$\mathbb{R}^ N$$ and applications.(English)Zbl 0856.35008

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.
Reviewer: I.Ginchev (Varna)

### MSC:

 35B10 Periodic solutions to PDEs 35K55 Nonlinear parabolic equations 35B35 Stability in context of PDEs

### Keywords:

Fischer’s equation