Parabolic Equations

Abstract

The problems we shall solve now are of the so-called heat equation type. Such theory is available in many places (see for instance [21], [2], [6], [17], [8]) but for the self-completeness of this book we would like to recall the main ideas. Φ being a domain of ℝ _n , we would like, for instance, to find a function u(x, t) such that

$$\left\{ \begin{gathered} {u_t} - \Delta u = f in \Omega \times (0,T), \hfill \\ u(x,t) = 0 on \Gamma \times (0,T), \hfill \\ u(x,0) = {u_0}(x) on \Omega , \hfill \\ \end{gathered} \right.$$

(8.1)

f(x,t), u ₀(x) are two given data. Clearly, the second equation of (8.1) can also be interpreted as

$$ u( \cdot ,t) \in H_0^1\left( \Omega \right). $$

(8.2)