Quasi-linear parabolic problems in L1 with non homogeneous conditions on the boundary

Abstract

We are interested in parabolic problems with L1 data of the type $$ (P_{i,j} (\phi ,\psi ,\beta ))\left\{ {\begin{array}{*{20}l} \delta _i u'(t) - {\text{div}}\,a(.,Du) = \phi (t){\text{ in }}Q: = (0,T) \times \Omega , \hfill \\ \delta _j u'(t) + a(.,Du).\eta + \beta (u) \mathrel\backepsilon \psi (t)\,{\text{on}}\;\sum := (0,T) \times \partial \Omega \hfill \\ \delta _i u(0,.) = u_0 {\text{ in }}\Omega , \hfill \\ \delta _j u(0,.) = \bar u_0 {\text{ on }}\delta \Omega , \hfill \end{array}} \right. $$ with i, j=0, 1, (i, j) ≠ (0, 0), δ0 = 0 and δ1 = 1. Here, Ω is an open bounded subset of \(\mathbb{R}^N \) with regular boundary ∂Ω and \(a:\Omega \times \mathbb{R}^N \to \mathbb{R}^N \) is a Caratheodory function satisfying the classical Leray-Lions conditions and β is a monotone graph in \(\mathbb{R}^2 \) with closed domain and such that \(0 \in \beta (0).\) We study these evolution problems from the point of view of semi-group theory, then we identify the generalized solution of the associated Cauchy problem with the entropy solution of \((P_{i,j} (\phi ,\psi ,\beta ))\) in the usual sense introduced in [5].