Strong Solutions of Doubly Nonlinear Parabolic Equations

Abstract

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

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