Spatial Behavior for Nonlinear Heat Equations

Abstract

In this paper we investigate spatial decay and growth estimates for the solutions of the nonlinear heat equation in a three-dimensional cylinder with homogeneous Dirichlet conditions prescribed on the lateral surface for all time. We derive two Phragmen–Lindelöf type growth–decay estimates. Two methods are used in our approximation. One of them involves a measure on the cross-sections and the other uses the "energy" contained in the part of the cylinder beyond a cross-section. For suitable volumetric heat capacity and thermal conductivitythe second-order approximation combined with comparison arguments allow us to improve the decay estimates. We also sketch the application of the first-order method in the case where the solid is a cone. Spatial estimates for the backward nonlinear equation are presented in the last section.