Topological characteristic of fully nonlinear parabolic boundary value problems

Igor V. Skrypnik, Igor B. Romanenko



A general nonlinear initial boundary value problem
\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
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.


Parabolic problems; nonlinear boundary condition; fully nonlinear parabolic equation; operator of type (S)$_+$; Galerkin approximations

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