A nonlinear degenerate diffusion equation not in divergence form

Abstract

We consider positive solution of the nonlinear degenerate diffusion equation \(u_t=u^p(\Delta u+u)\) with Dirichlet boundary condition and \(p>1\). It is proved that all positive solutions exist globally if and only if \(\lambda_1\ge 1\), where \(\lambda _1\) is the first eigenvalue of \(-\Delta\) on \(\Omega\) with homogeneous Dirichlet boundary condition.