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September/October 2011 Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions
Giuseppe Maria Coclite, Gisèle Ruiz Goldstein, Jerome A. Goldstein
Adv. Differential Equations 16(9/10): 895-916 (September/October 2011).

Abstract

Of concern is the nonlinear parabolic problem with nonlinear dynamic boundary conditions \begin{align*} & u_t +\,{\rm div}(F(u))=\,{\rm div}({\mathcal A}\nabla u),\qquad u(0,x)=f(x), \\ & u_t +\beta{{{\partial}^{\mathcal A}_{\nu}}} u+\gamma(x, u)-q\beta {\Delta_{\rm LB}} u=0, \end{align*} for $x\in \Omega\subset \mathbb R^N$ and $t\ge0$; the last equation holds on the boundary ${\partial}\Omega$. Here ${\mathcal A}=\{a_{ij}(x)\}_{ij}$ is a real, Hermitian, uniformly positive definite $N\times N$ matrix; $F\in C^1(\mathbb R^N;\mathbb R^N)$ is Lipschitz continuous; $\beta\in C({\partial}\Omega)$, with $\beta>0$; $\gamma:{\partial}\Omega\times\mathbb R\to \mathbb R; \,q\ge 0$; and ${{{\partial}^{\mathcal A}_{\nu}}} u$ is the conormal derivative of $u$ with respect to $A$; everything is sufficiently regular. Here we prove the well-posedness of the problem. Moreover, we prove explicit stability estimates of the solution $u$ with respect to the coefficients ${\mathcal A},$ $ F,$ $\beta,$ $\gamma,$ $q,$ and the initial condition $f$. Our estimates cover the singular case of a problem with $q=0$ which is approximated by problems with positive $q$.

Citation

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Giuseppe Maria Coclite. Gisèle Ruiz Goldstein. Jerome A. Goldstein. "Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions." Adv. Differential Equations 16 (9/10) 895 - 916, September/October 2011.

Information

Published: September/October 2011
First available in Project Euclid: 17 December 2012

zbMATH: 1231.35102
MathSciNet: MR2850757

Subjects:
Primary: 35K05, 35K60, 47H02

Rights: Copyright © 2011 Khayyam Publishing, Inc.

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Vol.16 • No. 9/10 • September/October 2011
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