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Zbl 0895.35050
Blanchard, D.; Murat, F.
Renormalised solutions of nonlinear parabolic problems with $L^1$ data: Existence and uniqueness.
(English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 127, No.6, 1137-1152 (1997). ISSN 0308-2105; ISSN 1473-7124/e

The authors prove an existence and uniqueness theorem for renormalized solutions to $${\partial\over \partial t} u-\text {div} a(t,x,Du) =f\text{ in } \Omega \times (0,t),$$ $$u|_{t=0} =u_0 \text{ in }\Omega, \quad u=0 \text{ on } \partial\Omega \times (0,T),$$ where $f$ and $u_0$ are $L^1$ functions on their domains of definition, and $a:(0,T) \times \Omega \times \bbfR^N\to \bbfR^N$ is a monotone (but not strictly monotone) Carathéodory function which defines a bounded, coercive, continuous operator on $L^p(0,T; W^{1,p}_0 (\Omega))$. A renormalized solution is a function $u\in C^0 ([0,T]$; $L^1 (\Omega))$ such that its truncates $T_K(u)\in L^p(0,T; W_0^{1,p} (\Omega))$, $$\lim_{K\to \infty} \int_{K\le | u|\le K+1} | Du|^p dxdt=0,$$ and it satisfies the differential equation in a generalized sense.
[R.Guenther (Corvallis)]
MSC 2000:
*35K60 (Nonlinear) BVP for (non)linear parabolic equations
35R05 PDE with discontinuous coefficients or data
35D05 Existence of generalized solutions of PDE

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