| Contents |
| We investigated the long-time behavior and solvability of the reaction-diffusion equation, which has a polynomial growth nonlinearity of arbitrary order, with Robin boundary condition on the bounded domain.
The problem that we investigate as the following:
\begin{equation*}
\left\{
\begin{array}{l}
u_{t}-\Delta u+a(x,t)\vert u\vert ^{\rho }u-b(x,t)\vert
u\vert ^{\nu }u=h(x,t),\text{ }(x,t)\in Q_{T} \ \ \ \ (1)\\(
\frac{\partial u}{\partial \eta }+k(x^{\prime },t)u)
\vert _{\partial \Omega }=\varphi (x^{\prime },t),\text{ }%
(x^{\prime },t)\in \Sigma_{T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) \\
u(x,0)=u_{0}(x) ,\text{ }%
x\in \Omega \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
\end{array}
\right.
\end{equation*}
here $\Omega \subset $ $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n}$($n\geq3$), is a bounded domain such that $\partial \Omega $
sufficiently smooth boundary, $T>0$, $Q_{T}=\Omega\times(0,T)$ and
$\Sigma_{T}=\partial\Omega\times[0,T]$.
In super linear case, for the existence and uniqueness of the generalized solution of problem (1)-(3), we obtain sufficient conditions for functions $a$, $b$ and $k$ and relations between $\rho$ and $\nu$ and under these conditions we show the existence of generalized solution of problem (1)-(3) and the uniqueness of the solution in corresponding spaces, by applying a general existence theorem.
For the long-time behavior, firstly we prove that solution has an absorbing set in
$L_{2}(\Omega)$. Secondly assuming that functions $h$ and $\varphi$ do not depend on the variable $t$, we prove the existence of an absorbing set in $W_{2}^{1}(\Omega )\cap L_{\rho+2}(\Omega)$. Also when the coefficients functions depend only on $x$ as well as $h$ and $\varphi$, we prove some asymptotic regularity and the existence of global attractor in $W_{2}^{1}(\Omega )\cap L_{\rho+2}(\Omega)$. |
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