Qualitative results for parabolic equations involving the p-Laplacian under dynamical boundary conditions

Abstract

We discuss comparison principles, the asymptotic behaviour, and the occurrence of blow up phenomena for nonlinear parabolic problems involving the \(p\)--Laplacian operator of the form \[ \left\{\begin{array}{ll} \partial_t u=\Delta_p u+f(t,x,u)&\mbox{in}\ \Omega\ \mbox{ for }\ t>0,\\ \sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0&\mbox{on}\ \partial\Omega\ \mbox{ for }\ t>0,\\ u(0,\cdot)=u_0 &\mbox{in}\ \overline{\Omega},\\ \end{array}\right. \] where \(\Omega\) is a bounded domain of \({\mathbb R}^N\) with Lipschitz boundary, and where \[\Delta_p u:={\mathrm div}\, \left(|\nabla u|^{p-2}\nabla u\right)\] is the \(p\)--{\itshape Laplacian} operator for \(p>1\). As for the dynamical time lateral boundary condition \(\sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0\) the coefficient \(\sigma\) is assumed to be a nonnegative constant. In particular, the asymptotic behaviour in the large for the parameter dependent nonlinearity \(f(\cdot,\cdot,u)=\lambda|u|^{q-2}u\) will be investigated by means of the evolution of associated norms.