The Nonlinear Case

Abstract

In this chapter we address elliptic and hyperbolic regularizations of the nonlinear parabolic problem introduced in Chapter 9 and denoted here P₀ :

$$ \left\{ \begin{gathered} u_t (x,t) - \Delta u(x,t) + \beta (u(x,t)) = f(x,t),(x,t) \in \Omega _T , \hfill \\ u(x,t) = 0{\text{ }}for (x,t) \in \Sigma _T , \hfill \\ u(x,0) = u_0 (x),x \in \Omega , \hfill \\ \end{gathered}\right. $$

where ω ⊂ ℝⁿ is a bounded open set with boundary ∂ω sufficiently smooth; T > 0 is a given time; $\Omega _T= \Omega\times (0,T];\Sigma _T= \partial \Omega\times \left[ {0,T} \right];f:\Omega _T\to \mathbb{R},u_0 :\Omega\to \mathbb{R}$ are given functions, f ∈ L²(ω_T), u₀ ∈ L²(ω).