Multiplicity and stability topics in semilinear parabolic equations

Abstract

Under suitable hypotheses on the nonlinear function $f$, the number of connected components of the complement of the nodal set of $\varphi$ is estimated when $\varphi$ is a solution of the elliptic equation $ -\Delta\varphi +f(\varphi) = 0$ in a bounded, open domain $\Omega$ with Dirichlet homogeneous boundary condition, and in the simplest case a dynamical consequence is derived for the corresponding semilinear heat equation. In addition, for simple domains such as a one-dimensional interval, a rectangle or a ball of arbitrary dimension, we establish the dynamical instability of solutions which do not have a constant sign in all the reasonable-looking cases.