Strong Solutions of Doubly Nonlinear Parabolic Equations

Abstract

The aim of this article is to discuss strong solutions of doubly nonlinear parabolic equations

where $A : X → X^*$ and $B : Y → Y^*$ are operators satisfying standard assumptions on boundedness, coercivity and monotonicity. Six different situations are identified which allow to prove the existence of a solution $u ∈ L^∞(0,T;X ∩ Y)$ to an initial value $u_0 ∈ X ∩ Y$, but only in some of these situations the equation is valid in a stronger space than $(X ∩ Y)^*$.