Behavior Near Time Infinity of Solutions of the Heat Equation

Abstract

Partial differential equations that include time derivatives of unknown functions are often called evolution equations. One important problem about evolution equations is to analyze the behavior of solutions at sufficiently large time. Such problems have been studied extensively from various points of view. Here, we are concerned with the initial value problem of the heat equation, which is a linear partial differential equation. It is not difficult to determine the asymptotic behavior of solutions of the heat equation near time infinity, and we introduce two methods to analyze its behavior. The first method is based on a representation formula of the solution of the equation directly; here we shall give a proof, which is short and easy. This method is sufficient to obtain the result for the heat equation; however, it may not apply to nonlinear problems in general, since we do not expect that solutions for nonlinear problems usually have a representation formula. The second method is based on a scaling transformation of the solution using the structure of the heat equation. By this method we shall give a proof of the behavior of solutions again. The proof by the second method is longer and it seems to be inefficient, but its idea can apply to nonlinear problems, which we study in Chapter 2 and in several parts of Chapter 3. To be familiar with the method, we give the proof for the heat equation, which is easier and more transparent to handle than nonlinear problems.