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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 $$u_t- \text{div}(A(t, x,\nabla u))= f\quad\text{in }]0, T[\times\Omega,\quad u= 0\quad\text{on }]0, T[\times\partial\Omega,\quad u(0,.)= u_0\quad\text{in }\Omega$$ with $u_0$ in $L^1(\Omega)$ and $f$ in $L^1(]0,T[\times \Omega)$ where $\Omega$ is an open bounded set of $\bbfR^N$ and $A$ is a Carathéodory function, satisfying some coercivity, monotonicity and growth conditions of Leray-Lions type, and defining an operator on $L^p(]0, T[; W^{1,p}_0(\Omega))$. 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 [{\it P. Bénilan}, {\it L. Boccardo}, {\it T. Gallouët}, {\it R. Gariepy}, {\it M. Pierre} and {\it 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
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References:
[1] ANDREU, F., MAZÓN, J. M., SEGURA DE LEÓN, S. and TOLEDO, J., Existence and uniqueness for a degenerate parabolic equation with l1-data, submitted. · Zbl 0912.35092
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