Global Existence and Decay Estimates for the Heat Equation with Exponential Nonlinearity

Abstract

In this paper, we consider the initial value problem in some Orlicz spaces for the heat equation with a nonlinearity having an exponential growth at infinity and vanishing at zero. Under some smallness conditions on the initial data and appropriate behavior near the origin for the nonlinearity, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time. We show in particular that the decay depends on the behavior of the nonlinearity near the origin.