In this chapter we study both the pure initial value problem and the mixed initial-boundary value problem for the model heat equation, using Fourier techniques as well as energy arguments. In Sect. 8.1 we analyze the solution of the pure initial value problem for the homogeneous heat equation by means of a representation in terms of the Gauss kernel, and use it to investigate properties of the solution. In the remainder of the chapter we consider the initial-boundary value problem in a bounded spatial domain. In Sect. 8.2 we solve the homogeneous equation by means of eigenfunction expansions, and apply Duhamel’s principle to find a solution of the inhomogeneous equation. In Sect. 8.3 we introduce the variational formulation of the problem and give examples of the use of energy arguments, and in Sect. 8.4 we show and apply the maximum principle.
KeywordsMaximum Principle Parabolic Equation Variational Formulation Heat Equation Parabolic Problem
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