Strong solutions of doubly nonlinear parabolic equations (Q415187)

Summary: The aim of this article is to discuss strong solutions of doubly nonlinear parabolic equations \[ \frac{\partial Bu}{\partial t} + Au = f, \] where \(A : X \rightarrow X^*\) and \(B : Y \rightarrow Y^*\) are operators satisfying standard assumptions on boundedness, coercivity and monotonicity. Six different situations are identified which allow to prove the existence of a solution \(u \in L^\infty (0,T;X \cap Y)\) 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 \((X \cap Y)^*\).