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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

tu-diva(t,x,Du)=finΩ×(0,t),
u| t=0 =u 0 inΩ,u=0onΩ×(0,T),

where f and u 0 are L 1 functions on their domains of definition, and a:(0,T)×Ω× N N is a monotone (but not strictly monotone) Carath√©odory function which defines a bounded, coercive, continuous operator on L p (0,T;W 0 1,p (Ω)). A renormalized solution is a function uC 0 ([0,T]; L 1 (Ω)) such that its truncates T K (u)L p (0,T;W 0 1,p (Ω)),

lim K K|u|K+1 |Du| p dxdt=0,

and it satisfies the differential equation in a generalized sense.

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35R05PDEs with discontinuous coefficients or data
35D05Existence of generalized solutions of PDE (MSC2000)