The Decay of the Solutions for the Heat Equation with a Potential

We Study the large time behavior of the solutions for the Cauchy problem, partial derivative(t)u = Delta u + a(x,t)u in R-N x (0, infinity), u(x,0) = phi(x) in R-N, where phi is an element of L-1(R-N, (1+|x|(K))dx) with K >= 0 and parallel to a(t)parallel to(L infinity(RN)) = O(t(-A)) as t -> infinity for some A > 1. In this paper we classify the decay rate of the solutions and give the precise estimates on the difference between the solutions and their asymptotic profiles. Furthermore, as an application, we discuss the large time behavior of the global solutions for the semilinear heat equation, partial derivative(t)u = Delta u + lambda|u|(p-1)u, where lambda is an element of R and p > 1.