Renormalized solutions of doubly nonlinear parabolic equations. (Solutions renormalisées d’équations paraboliques à deux non linéarités.) (French. Abridged English version) Zbl 0810.35038

Results about existence, uniqueness and comparison are established for renormalized solutions of the following class of doubly nonlinear parabolic initial boundary value problems: \[ \partial_ t b(u)- \Delta u+ \text{div } \Phi(u)=f \quad \text{in } \Omega\times (0,T), \]
\[ u=0 \quad \text{on } \partial\Omega\times (0,T), \qquad b(u)|_{t=0} =b(u_ 0) \quad \text{in } \Omega. \] Here \(\Omega\subset \mathbb{R}^ N\) is bounded and open, \(b\) is a strictly increasing \(C^ 1\)-function on an open interval \(I\) with \(b(I)= \mathbb{R}\) and \(b(0)=0\), \(b^{-1}: \mathbb{R}\to I\) and \(\Phi: \mathbb{R}\to \mathbb{R}^ N\) are continuous, \(u_ 0\) is a measurable function such that \(b(u_ 0)\in L^ 1(\Omega)\), and \(f\in L^ 1(\Omega\times (0,T))\).
Reviewer: L.Recke (Berlin)


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)