Linear Parabolic Equations

Abstract

The problems we shall solve now are of the so-called heat equation type (see chap. 1). That is to say, Ω being a domain of ℝⁿ, we would like, for instance, to find a function u(x, t) such that

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

(11.1)

f(x, t), u₀(x) are two given data. A strong solution to (11.1) could be a function u such that all the above equalities hold in a usual sense. Clearly — as we did already for the Dirichlet problem — the second equation of (11.1) can also be interpreted as

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

(11.2)