## 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.

### MSC:

 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)
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### References:

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