DECAY BOUNDS FOR SOLUTIONS OF 2ND-ORDER PARABOLIC PROBLEMS AND THEIR DERIVATIVES

For a class of quasilinear parabolic equations, we derive in this pape r a new maximum principle for the derivatives of solutions of homogene ous Dirichlet and Neumann initial boundary value problems. In the line ar heat equation this principle is used to derive new explicit decay b ounds for the absolute value of the gradient of the solution, and in t he quasilinear problems, criteria are determined which imply that solu tions decay exponentially. Explicit decay bounds are then established.