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 Title Solvability and long time behavior of nonlinear Reaction-Diffusion equations with Robin Boundary Condition
 Name Eylem Ozturk Country Turkey Email eyturk1983@gmail.com Co-Author(s) Kamal Soltanov Submit Time 2014-02-23 06:27:06 Session Special Session 37: Global or/and blowup solutions for nonlinear evolution equations and their applications
 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 $W_{2}^{1}(\Omega )\cap L_{\rho+2}(\Omega)$.