Existence and convergence results for evolution hemivariational inequalities (Q5929990)

Let \(V\) and \(X\) be two reflexive separable Banach spaces where \(V\) is dense in \(X\) and consider \({\mathcal V}=L^p(0,T;V)\), \({\mathcal X}=L^p(0,T;X)\), \(2\leq p<\infty\), \(0<T<\infty\).NEWLINENEWLINENEWLINEIn this framework, the author gives two main results.NEWLINENEWLINENEWLINEFirst, it is considered the evolution hemivariational inequality NEWLINE\[NEWLINE\left<u'+Au-f,v-u\right>_{\mathcal V} + {\mathcal J}^0(u;v-u) \geq 0 \text{ for all } v\in {\mathcal V}NEWLINE\]NEWLINE where \(A:{\mathcal V}\to 2^{{\mathcal V}^*}\) is a nonlinear multivalued operator which is bounded, coercive and generalized pseudomonotone, \(f\in {\mathcal V}^*\), \({\mathcal J}\) is a locally Lipschitz functional defined on \({\mathcal X}\) and \({\mathcal J}^0\) is its generalized directional derivative at \(u\). The derivative \(u'\) is understood in the sense of vector-valued distributions. The problem can be formulated as the abstract evolution inclusion NEWLINE\[NEWLINEu'+Au+\partial{\mathcal J}(u)\ni f, \quad u(0)=0,NEWLINE\]NEWLINE where \(\partial{\mathcal J}\) is the generalized gradient in the sense of Clarke. The existence of solutions \(u\) such that \(u'\in {\mathcal V}^*\) is provided. As a consequence, an existence result on a problem with a nonzero initial data and single-valued operator \(A\) is given.NEWLINENEWLINENEWLINEThen, problems NEWLINE\[NEWLINEu'_h+A_h u_h+\partial{\mathcal J}_h(u_h)\ni f_h,\quad u_h(0)=u_0^h, \quad h\in \mathbb N,NEWLINE\]NEWLINE are considered, where \(A_h\) are in correspondence with single-valued nonlinear operators of divergence form. A convergence result is given, concerning the upper semicontinuity of the solution set to the parabolic hemivariational inequalities.