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Renormalised solutions of nonlinear parabolic problems with \(L^1\) data: Existence and uniqueness. (English) Zbl 0895.35050
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 \(a:(0,T) \times \Omega \times \mathbb{R}^N\to \mathbb{R}^N\) 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 \(T_K(u)\in L^p(0,T; W_0^{1,p} (\Omega))\), \[ \lim_{K\to \infty} \int_{K\leq | u|\leq K+1} | Du|^p dxdt=0, \] and it satisfies the differential equation in a generalized sense.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
35D05 Existence of generalized solutions of PDE (MSC2000)
Full Text: DOI
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