Solvability of nonautonomous parabolic variational inequalities in Banach spaces

Abstract

We consider nonautonomous parabolic variational inequalities having the strong formulation \begin{align*} \langle { u'(t) }{ v - u(t) } \rangle + \langle { A(t) u(t) }{ v - u(t) }\rangle + \Phi(t, v ) - \Phi(t,u(t)) \geq 0, \\ \forall \, v \in V^{**}, \ a.e. \ t \geq s, \end{align*} where $u(s) = u_{s}$ for some admissible initial datum, $V$ is a separable Banach space with separable dual $V^{*}$, $A(t) : V^{**} \rightarrow V^{*}$ is an appropriate family of monotone operators, and $\Phi(t,\cdot) : V^{**} \rightarrow \mathbb R \cup \{ \infty \}$ is a family of convex, weak* lower-semicontinuous functionals. Well-posedness follows from an explicit construction of the related evolution family $\{ U(t,s) : t \geq s \}$. Illustrative applications are given.