Topological characteristic of fully nonlinear parabolic boundary value problems

A general nonlinear initial boundary value problem $$ \align \frac{\partial u}{\partial t} - F(x,t,u,D^{1}u,\dots, D^{2m}u)&=f(x,t), \tag 1 \\ &\hskip -30pt (x,t)\in Q_{T}\equiv \Omega\times (0,T), \\ G_{j}(x,t,u,\dots, D^{m_{j}}u)&=g_{j}(x,t), \tag 2\\ &\hskip-30pt (x,t)\in S_{T}\equiv \partial\Omega\times (0,T), j=\overline{1,m}, \\ u(x,0)=h(x),\quad& x\in\Omega \tag 3 \endalign $$ is being considered, where $\Omega$ is a bounded open set in $\R^n$ with sufficiently smooth boundary. The problem (1)-(3) is then reduced to an operator equation $Au=0$, where the operator $A$ satisfies (S)$_+$ condition. The local and global solvability of the problem (1)-(3) are achieved via topological methods developed by the first author. Further applications involving the convergence of Galerkin approximations are also given