Quasi-linear parabolic problems in L 1 with non homogeneous conditions on the boundary期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Quasi-linear parabolic problems in L 1 with non homogeneous conditions on the boundary

Authors:

Institution:

(1) TU Berlin, Fakultaet II-Mathematik und Naturwisseuschaften, Institut fuer Mathematik, Sekretariat 6-3, Strasse des 17 juni 136, 10623 Berlin, Germany

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),

₀ = 0 and

₁ = 1. Here, OHgr

is an open bounded subset of $\mathbb{R}^N$ with regular boundary part

and $a:\Omega \times \mathbb{R}^N \to \mathbb{R}^N$ is a Caratheodory function satisfying the classical Leray-Lions conditions and beta

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].

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35K55 35K60 35D05

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