Self-similar subsolutions and blow-up for nonlinear parabolic equations

Abstract

For a wide class of nonlinear parabolic equations of the form u t -Δu = F(u,▽u ), we give finite time blow-up results for large initial data. Blow-up time estimates are also provided. These results rely on a new method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results on reaction-diffusion equations, on generalized Burgers' equations and on the equation of Chipot and Weissler. 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.