Regularity for nonlinear variational evolution inequalities in Hilbert spaces (Q2766292)

Let \(H\) and \(V\) be two real separable Hilbert spaces such that \(V\) is a dense subspace of \(H\). Let the single-valued operator \(A\) be given which is hemicontinuous and coercive from \(V\) to \(V^{*}\). Here, \(V^{*}\) stands for the dual space of \(V\). Let \(\varphi: V\to (-\infty,+\infty]\) be a lower semicontinuous proper convex function. The authors study the existence, uniqueness and a variation of solution to the following initial value problem NEWLINE\[NEWLINE \frac{d x(t)}{d t}+A x(t)+\partial \varphi (x(t))\notin f(t,x(t))+h(t), \quad t\in [0,T],NEWLINE\]NEWLINE \( x(0)=x_{0},\) where \(f:\mathbb{R}\times V\to H\) is Lipschitz continuous, \(h:\mathbb{R}\to H\) and \(\partial \varphi: V\to V^{*}\) is the subdifferential multivalued operator of \(\varphi\) defined by NEWLINE\[NEWLINE \partial\varphi (x)=\{x^{*}\in V^{*}: \varphi(x)\leq \varphi(y) + (x^{*},x-y), \quad y\in V \},NEWLINE\]NEWLINE where \((.,.)\) denotes the duality pairing between \(V^{*}\) and \(V\).NEWLINENEWLINEFor the entire collection see [Zbl 0973.00042].