Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Abstract

Consider the Dirichlet problem for the parabolic equation \(u_t=\Delta u+f(x,t,u)\) in \(\Omega \times(0,\infty)\), where $\Omega$ is a bounded domain in \(\mathbb{R}^n\) and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form \(f(x,t,u)=m(t)g(u)\) where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.