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

Alessio Fiscella; Enzo Vitillaro

doi:10.3934/dcds.2013.33.5015

Article Contents

2013, Volume 33, Issue 11&12: 5015-5047. Doi: 10.3934/dcds.2013.33.5015

This issue Previous Article Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations Next Article On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces

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

Alessio Fiscella^1, and
Enzo Vitillaro^2,

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

Received: September 2011

Published: November 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K20, 35B44; Secondary: 35K57, 35Q79, 35K58.

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