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

Abstract

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