Solvability problem for strong-nonlinear nondiagonal parabolic system

summary:A class of $q$-nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities $q\in (1,2)$, $q=2$, $q>2$, are analyzed