Semi linear parabolic equations with nonlinear general Wentzell boundary conditions

Mahamadi Warma

doi:10.3934/dcds.2013.33.5493

Article Contents

2013, Volume 33, Issue 11&12: 5493-5506. Doi: 10.3934/dcds.2013.33.5493

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Semi linear parabolic equations with nonlinear general Wentzell boundary conditions

Mahamadi Warma^1,

1.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received: September 30, 2011

Revised: September 30, 2012

Published: October 2013

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Abstract

Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.

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Mathematics Subject Classification: 35J20, 47H20.

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