Weak and strong solvability of parabolic variational inequalities in Banach spaces

Abstract.

We consider parabolic variational inequalities having the strong formulation

$$ \left\{ {\begin{array}{*{20}c} {\left\langle {u'(t),\,v - \left. {u(t)} \right\rangle + \left\langle {Au(t),} \right.\,v - \left. {u(t)} \right\rangle + \Phi (v) - \Phi (u(t) \geq 0,} \right.} \\ {\forall v \in V^{**} ,\,a.e.\,t \geq 0,} \\ \end{array} } \right. $$

((1))

where $u(0) = u_0$ for some admissible initial datum, V is a separable Banach space with separable dual $V^* ,A:V^{**} \to V^*$ is an appropriate monotone operator, and $\Phi :V^{**} \to \mathbb{R} \cup \{ \infty \}$ is a convex, ${\text{weak}}^*$ lower semicontinuous functional. Well-posedness of (1) follows from an explicit construction of the related semigroup $\{ S(t):t \geq 0\} .$ Illustrative applications to free boundary problems and to parabolic problems in Orlicz-Sobolev spaces are given.