| Contents |
| We study some third type boundary value problems for a general semilinear parabolic equation in divergence form:
\begin{equation}
\frac{\partial u}{\partial t}+Lu+g(x,t,u)=h(x,t),\text{ }(x,t)\in
Q_{T}\equiv \Omega \times (0,T] \tag{1}
\end{equation}
\begin{equation}
u(x,0)=u_{0}(x),\text{ } x\in\Omega\subset%
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%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}, n\geq3 \tag{2}
\end{equation}
\begin{equation}
\left. \left( \frac{\partial u}{\partial \nu }+k(x,t)u\right)
\right\vert _{\Gamma_{T}}=\varphi (x,t),\text{ } \Gamma
_{T}\equiv
\partial \Omega \times \lbrack 0,T], T>0 \tag{3}
\end{equation}
Here $\Omega $ is a bounded domain with sufficiently smooth boundary $\partial\Omega$; $L$ denotes a second order linear elliptic operator in divergence form:
\begin{equation*}
Lu:=-\overset{n}{\underset{i,j=1}{\sum }}D_{i}(a_{ij}\left( x,t\right) D_{j}u)+%
\overset{n}{\underset{i=1}{\sum }}b_{i}\left(x,t\right)
D_{i}u+c(x,t)u,
\end{equation*}
where $a_{ij}$, $b_{i}$ and $c$ are given coefficient functions $\left( i,j=1,...,n\right) $;
$g:Q_{T} \times
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\mathbb{R}
%EndExpansion
\longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $k:\Gamma_{T} \longrightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ are given functions;
$h$ and $\varphi $ are given generalized functions.
For the existence and the uniqueness of the generalized solution of problem (1)-(3), we obtain sufficient conditions for $L,g$ and $k$. Under these conditions we prove that problem (1)-(3) is uniquely solvable in corresponding spaces by applying a general
existence theorem.
For the long time behavior of solution, we obtained the existence of the absorbing sets in two different spaces for the autonomous case of the problem. |
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