Decay Bounds for Solutions of Second Order Parabolic Problems and Their Derivatives

This is the authors' abstract. We don't add key points for this paper.

For a class of quasilinear parabolic equations, we derive in this paper a new maximum principle for the derivatives of solutions of homogeneous Dirichlet and Neumann initial boundary value problems. In the linear heat equation this principle is used to derive new explicit decay bounds for the absolute value of the gradient of the solution, and in the quasilinear problems, criteria are determined which imply that solutions decay exponentially. Explicit decay bounds are then established.