Spatial behaviour for quasilinear parabolic equations in cylinders and cones

Abstract

In this paper we investigate spatial decay and growth estimates for the solutions of a quasilinear parabolic equation defined in a three dimensional cylinder with homogeneous Dirichlet condition prescribed on the lateral surface for any time. We derive a Phragmen-Lindelof type growth-decay estimate. An upper bound for the “energy” contained in the whole cylinder is also obtained. In the last section we sketch the extension of the method for conical domains. We use the energy method for the class of equations which satisfy a certain condition.