Self-Similar Subsolutions and Blowup for Nonlinear Parabolic Equations

Abstract

For a wide class of nonlinear parabolic equations of the formut−Δu=F(u,∇u), we prove the nonexistence of global solutions for large initial data. We also estimate the maximal existence time. To do so we use a method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results onut−Δu=up, on generalized Burgers' equations, and on other semilinear equations. This method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems, both semilinear and quasilinear.