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| The talk will deals with local well--posedness, global existence and blow--up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is
$$u_{t}-\Delta u=\left|u\right|^{p-2}u\qquad\mbox{in}\quad \left(0,\infty\right)\times\Omega,$$
$$u=0\qquad\mbox{on}\quad \left[0,\infty\right)\times\Gamma_{0},$$
$$\frac{\partial u}{\partial\nu} =-\left|u_{t}\right|^{m-2}u_{t}\qquad\mbox{on}\quad \left[0,\infty\right)\times\Gamma_{1},$$
$$u\left(0,x\right)=u_{0}\left(x\right)\qquad\mbox{in}\quad \Omega.$$
where $\Omega$ is a bounded open regular domain of ${I\!\!R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ and $u_0\in H^1(\Omega)$, ${u_0}_{|\Gamma_0}=0$.After showing local well--posedness in the Hadamard sense we give global existence and blow--up results when $\Gamma_0$ has positivesurface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms $|u|^{p-2}u$ and $|u_{t}|^{m-2}u_{t}$ are respectively replaced by $f\left(x,u\right)$ and $Q(t,x,u_t)$ under suitable assumptions on
them. |
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