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Zeitschrift für Analysis und ihre Anwendungen

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Volume 31, Issue 2, 2012, pp. 217–235
DOI: 10.4171/ZAA/1456

Published online: 2012-03-30

Strong Solutions of Doubly Nonlinear Parabolic Equations

Aleš Matas[1] and Jochen Merker[2]

(1) University of West-Bohemia, Pilsen, Czech Republic
(2) Universität Rostock, Germany

The aim of this article is to discuss strong solutions of doubly nonlinear parabolic equations $$\frac{\partial Bu}{\partial t} + Au = f,$$ where $A : X → X^*$ 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 $(X ∩ Y)^*$.

Keywords: Doubly nonlinear evolution equations, elliptic-parabolic problems, strong solutions, regularity

Matas Aleš, Merker Jochen: Strong Solutions of Doubly Nonlinear Parabolic Equations. Z. Anal. Anwend. 31 (2012), 217-235. doi: 10.4171/ZAA/1456