Parabolic variational inequalities and their Lipschitz perturbations in Hilbert spaces (Q1585259)

This article deals with the Cauchy problem \[ {dx(t)\over dt}+ Ax(t)+\partial\phi(x(t))\ni f(x(t))+ k(t)\quad (0< t\leq T),\quad x(0)= x_0, \] where \(A\) is a linear (nonmonotone) operator which generates an analytic semigroup and such that \[ (Au, u)\geq \omega_1\|u\|_V- \omega_2\|u\|_H\qquad (\omega_1,\omega_2\in \mathbb{R}), \] \(\partial\phi: V\to V^*\) is a subdifferential operator of a lower semicontinuous, proper convex function \(\phi: H\to (-\infty,\infty)\), \(H\) and \(V\) are Hilbert spaces, \(V\) is continuously imbedded in \(H\), \(f(x): V\to H\) is a Lipschitzian function, \(k(t)\in L^2(0,T; H)\). Described are conditions under which the problem has a solution \(x(t)\in L^2(0,T;V)\cap C([0, T]; H)\) such that \(\|x\|_{L^2\cap C}\leq c(1+\|x_0\|_V+ \|k\|_{L^2(0, T;H)})\) or even \(x(t)\in L^2(0, T;D(A))\cap W^{1,2}(0, T;H)\cap C([0, T], H)\) such that \(\|x\|_{L^2\cap W^{1,2}\cap C}\leq c(1+\|x_0\|_V +\|k\|_{L^2(0, T;H)})\) and the mapping \((x_0, k)\mapsto x\) is continuous between spaces \(V\times L^2(0,T; H)\) and \(L^2(0, T;D(A))\cap W^{1,2}(0, T; H)\).