A nonlinear degenerate diffusion equation not in divergence form
The interesting paper under review studies positive solutions of the nonlinear degenerate diffusion problem: \[ u_t=u^p(\Delta u+u), \quad x\in \Omega,\;t>0,\qquad u=0, \quad x\in \partial\Omega,\;t>0,\qquad u(x,0)=\phi(x), \quad x\in \Omega, \] where \(\Omega\subset {\mathbb R}^N\) is a bounded domain and \(p>1.\) The authors prove that all positive classical solutions to the above problem exist globally if and only if \(\lambda_1\geq 1\) where \(\lambda_1\) is the first eigenvalue of \(-\Delta \) in \(\Omega\) with homogeneous Dirichlet boundary data.
scientific article; zbMATH DE number 1437372