A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear ( but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases. One advantage of our approach is that the parabolic semiflow is naturally order preserving, in contrast to pseudo- gradient flows considered when using variational methods. Therefore we can obtain much information on nodal properties of equilibria that was not known before.