The Heat Equation

Abstract

This chapter is devoted to the equation

$$\frac{{\partial u}}{{\partial t}}\left( {x,t} \right) - {\omega ^2}\Delta u\left( {x,t} \right) = f\left( {x,t} \right),x \in \Omega \subset {R^n},t \in \left( {0,T} \right)$$

where Δ is the Laplace operator with respect to the spatial variable x and t is the time variable. This equation is called the the heat equation and as the name suggests, it is relevant in the mathematical models of heat conduction and of diffusion processes. Primarily, we shall be concerned with existence of solutions to the boundary value problem on bounded domains and with the Cauchy problem in R ⁿ. The treatment is classical (the Fourier method) but the existence results are obtained in the framework of Sobolev spaces. Moreover, in the last section we briefly describe an alternative method to these problems, the semigroup approach, which has the advantage to be applicable to a broad class of boundary value problems associated with evolutionary partial differential equations.