Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness

D. Blancharda1 and F. Murata2

a1 URA-CNRS 1378—Analyse et Modèles Stochastiques, Université de Rouen, 76821 Mont Saint Aignan cedex, France

a2 URA-CNRS 189—Laboratoire d'Analyse Numérique, Tour 55–65, Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France

In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem

S0308210500026986_eqnU1

where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) × Ω × ℝN → ℝN is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C0 ([ 0, T] L1 (Ω)) such that its truncates TK(u) belong to with

S0308210500026986_eqnU2

this solution satisfies the equation formally obtained by using in the equation the test function S(u)φ, where φ belongs to and where S belongs to C(ℝ) with

(Received July 10 1996)