ASYMPTOTIC BEHAVIOR FOR TWO REGULARIZATIONS OF THE CAUCHY PROBLEM FOR THE BACKWARD HEAT EQUATION

Abstract

One method of regularizing the initial value problem for the backward heat equation involves replacing the equation by a singularly perturbed hyperbolic equation which is equivalent to a damped wave equation with negative damping. Another regularization of this problem is obtained by perturbing the initial condition rather than the differential equation. For both of these problems, we investigate the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus establish spatial decay results of the Saint-Venant type.