Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Abstract

Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as t → ∞ of the L 1 -norm of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.