Positive unstable periodic solutions for superlinear parabolic equations

Abstract:

In this paper, we are concerned with a superlinear parabolic equation \[ \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}}{{\partial t}} - \Delta u = {u^p} + h(t,x),} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \Omega ,} \hfill \\ {{u = 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ {{u > 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ \end {array} } \right .\] where $\Omega \subset {{\mathbf {R}}^N}$ is a bounded domain with smooth boundary $\partial \Omega$, h is T-periodic with respect to the first variable, and $1 < p < \frac {{N + 2}}{{N - 2}}$ if $N \geq 3$ and $1 < p < + \infty$ if $N \leq 2$. It is shown that there exist a stable and an unstable positive T-periodic solution for this problem if h is sufficiently small in ${L^\infty }$.