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November  2013, 33(11&12): 5015-5047. doi: 10.3934/dcds.2013.33.5015

## Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions

 1 Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy 2 Dipartimento di Matematica ed Informatica, Università di Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy

Received  September 2011 Published  May 2013

The paper 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 $\begin{cases} u_{t}-\Delta u=|u|^{p-2} u in (0,\infty)\times\Omega,\\ u=0 on [0,\infty) \times \Gamma_{0},\\ \frac{\partial u}{\partial\nu} = -|u_{t}|^{m-2}u_{t} on [0,\infty)\times\Gamma_{1},\\ u(0,x)=u_{0}(x) in \Omega \end{cases}$ where $\Omega$ is a bounded open regular domain of $\mathbb{R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ 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 positive surface 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.
Citation: Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015
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