Renormalised solutions of nonlinear parabolic problems with <i>L</i>¹ data: existence and uniqueness

Abstract

In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to L 1 (Ω × (0, T )) and L 1 (Ω), and where the function a :(0, T ) × Ω × ℝ N → ℝ N 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 C 0 ([ 0, T ] L 1 (Ω)) such that its truncates T K (u) belong to with 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