Existence and Meyers estimate for nonlinear parabolic variational inequalities (Q1124051)

The paper deals with nonlinear parabolic variational inequalities \[ \int^{t}_{0}(<D_ tv,\Phi \sigma '(v-u)>_{H^{-1},H^ 1_ 0}+<A(.,u)u,\quad \Phi \sigma '(v-u)>_{H^{-1},H^ 1_ 0})ds+ \] \[ +\int^{t}_{0}\int_{\Omega}H(x,t,u,D_ xu)\Phi \sigma '(v-u)dx ds+\int^{t}_{0}\int_{\Omega}f_ 0\Phi \sigma '(v-u)dx ds+ \] \[ +\sum^{N}_{i=1}\int^{t}_{0}\int_{\Omega}f_ iD_{x_ i}(\Phi \sigma '(v-u))dx ds+\int^{t}_{0}\int_{\Omega}D_ t\Phi \sigma (v- u)dx ds- \] \[ -\| \Phi \sigma (v-u)\|_{L^ 1(\Omega)}(t)+\| \Phi \sigma (v-u_ 0)\|_{L^ 1(\Omega)}(0)\geq 0, \] \(\forall \sigma \in C^ 2(R)\), \(\sigma\) convex, \(\sigma '(0)=0\), \(\Phi \in C^{\infty}(\bar Q)\), \(\Phi \geq 0,\) \(\forall v\in H^ 1((0,T);H^{-1}(\Omega))\cap L^ 2((0,T);H^ 1_ 0(\Omega))\cap L^{\infty}(Q)\), \(v\leq \psi\) a.e. in Q, \(with\) u\(\in C(0,T;L^ 2(\Omega))\cap L^ 2(0,T;H^ 1_ 0(\Omega))\cap L^{\infty}(\Omega)\), \(u\leq \psi\) a.e. in Q, \(u(0)=u_ 0,\) where the operator H is of quadratic growth in the gradient. Using the estimate \(\phi\leq u\leq \sup_{Q}\psi\), where \(\phi\) is a subsolution of the above problem, the author proves existence and regularity theorems for a solution u.