Various Properties of Solutions of the Heat Equation

Abstract

Here we establish the tools used in Chapter 1 in order to analyze the asymptotic behavior of solutions for the heat equation. We start by deriving L ^p-L ^q estimates for solutions and their derivatives and the uniqueness theorem for weak solutions. For this purpose, we prepare the Young inequality for convolution, which has a wide range of applications. Furthermore, algebraic and commutativity properties, in particular concerning differentiation of convolutions, are stated. These properties turn out to be helpful in the proof of smoothness for t > 0 for the solution of the heat equation in Chapter 1. Next, we consider the continuity of the solution at time t= 0, in the case that the initial value is continuous. Continuity is proved by a fairly general method that applies to a large class of equations.