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Zbl 0909.35075
Prignet, Alain
Existence and uniqueness of ``entropy'' solutions of parabolic problems with $L^1$ data. (English)
[J] Nonlinear Anal., Theory Methods Appl. 28, No.12, 1943-1954 (1997). ISSN 0362-546X

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 $L^p(]0, T[; W^{1,p}_0(\Omega))$.\par 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 2000:: *35K60 (Nonlinear) BVP for (non)linear parabolic equations
35D05 Existence of generalized solutions of PDE
35R05 PDE with discontinuous coefficients or data

Keywords: nonlinear parabolic equation; existence uniqueness

Citations: Zbl 0866.35037

Cited in: Zbl 1019.35049

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