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The Nonlinear Case

Part of the International Series of Numerical Mathematics book series (ISNM, volume 156)

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, fL2T), u0L2(ω).

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Copyright information

© Birkhäuser Verlag AG 2007

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