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

Consider the Dirichlet problem for the parabolic equation in , where$Omega$ is a bounded domain in and f has superlinear subcritical growth in u.If f is independent of t and satisfies someadditional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivialstationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technicalconditions then we prove the existence of a positive periodic solution andwe provide a locally uniform bound for all global solutions.