Abstract
A general nonlinear initial boundary value problem \begin{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*{$(3)$} \end{align} is being considered, where $\Omega$ is a bounded open set in $\mathbb R^n$ 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.
Citation
Igor V. Skrypnik. Igor B. Romanenko. "Topological characteristic of fully nonlinear parabolic boundary value problems." Topol. Methods Nonlinear Anal. 23 (1) 1 - 31, 2004.
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