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Existence and uniqueness of “entropy” solutions of parabolic problems with L 1 data. (English) Zbl 0909.35075

From the introduction: The author solves the parabolic equation

u t -div(A(t,x,u))=fin]0,T[×Ω,u=0on]0,T[×Ω,u(0,·)=u 0 inΩ

with u 0 in L 1 (Ω) and f in L 1 (]0,T[×Ω) where Ω is an open bounded set of N and A is a Carathéodory function, satisfying some coercivity, monotonicity and growth conditions of Leray-Lions type, and defining an operator on L p (]0,T[;W 0 1,p (Ω)).

In order to obtain an existence uniqueness result, an entropy formulation is proposed, which is very close to the one which has been introduced for the elliptic case in [P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vasquez, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, 241-273 (1995; Zbl 0866.35037)].


MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35D05Existence of generalized solutions of PDE (MSC2000)
35R05PDEs with discontinuous coefficients or data