### Topological characteristic of fully nonlinear parabolic boundary value problems

DOI: http://dx.doi.org/10.12775/TMNA.2004.001

#### Abstract

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.

$$

\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.

#### Keywords

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

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