An existence theorem for evolution inclusions involving opposite monotonicities (Q1269046)

Extending to separable Hilbert spaces previous results of \textit{M. Otani} [J. Fac. Sci. Univ. Tokyo Sect. IA 24, 575-605 (1977; Zbl 0386.47040)] and of \textit{A. Cellina} and \textit{V. Staicu} [J. Differ. Equations 90, No. 1, 71-80 (1991; Zbl 0719.34030)], the authors prove a theorem stating the existence of a strong global solution \(x(.):T:=[0,b]\to H\) to the differential inclusion: \[ -x'(t)\in \partial \phi (x(t)) - F(x(t)) \quad\text{a.e. }(T), \quad x(0)=x_0, \] in the case the subdifferential \(\partial \phi (.)\) of the proper convex l.s.c function \(\phi (.):H \to \overline \mathbb{R}\) and the multifunction \(F(.)\), whose values are nonempty closed subsets of the separable Hilbert space \(H\), satisfy certain regularity and growth conditions. The main result is used to obtain the existence of a ``state-control'' pair \[ (x(.),u(.))\in C(T,L^2(Z))\times L^2(Z) \] satisfying: \[ {{\partial x}\over {\partial t}} - \sum_{k=1}^{N}D_k(| D_kx| ^{p-2}D_kx)+u(z)=0 \quad\text{a.e. }(T\times Z), \] \[ x| T\times \Gamma =0, \quad x(0,z)=x_0(z), \quad u(z)\in \text{sgn}(x(z)), \] where \(Z \subset \mathbb{R}^N\) is a bounded domain with smooth boundary \(\Gamma\) and \(p\geq 2\).