Multivalued perturbations of subdifferential type evolution equations in Hilbert spaces (Q1117389)

The authors study the following perturbed evolution inclusion driven by a maximal multivalued operator of the subdifferential type, \[ (*)\quad - \dot x(t)\in \partial \phi (x(t))+F(t,(x(t)),\quad x(0)=x_ 0, \] \(t\in [0,b]\neq \emptyset\), \(b<\infty\). The mapping \(\phi\) is a proper, convex, lower semicontinuous function from a separable Hilbert space \({\mathbb{X}}\) into \(\overline{{\mathbb{R}}}\), and \(x_ 0\in {\mathbb{D}}(\partial \phi)=\{x\in {\mathbb{X}}:\) \(\partial \phi (x)\neq \emptyset \}\). For the case that F(.,.) is convex valued but not closed valued the authors prove an existence result, another one is shown for F(.,.) being nonconvex valued. A stability result for the evolution inclusion under consideration is obtained. After that a random version of (*) is introduced and investigated. Finally, the authors give some applications of their results to problems in partial differential equations.