Advances in Differential Equations

The heat equation with nonlinear general Wentzell boundary condition

Angelo Favini, Jerome A. Goldstein, Silvia Romanelli, and Gisèle Ruiz Goldstein

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Let $\Omega$ be a bounded subset of $\mathbf R^N$ with a $C^{2,\varepsilon}$ boundary $\partial\Omega$, $\alpha\in C^2(\overline\Omega)$ with $\alpha>0$ in $\overline{\Omega}$ and $A$ the operator defined by $Au:=\nabla\cdot (\alpha\nabla u)$ with the nonlinear general Wentzell boundary condition $$ Au+b\frac{\partial u}{\partial n}\in c\, \beta(\cdot, u)\quad\text{on}\quad\partial \Omega, $$ where $n(x)$ is the unit outer normal at $x$, $b,\, c$ are real-valued functions in $C^1(\partial\Omega)$ and $\beta (x,\cdot)$ is a maximal monotone graph. Then, under additional assumptions on $b, c, \beta$, we prove the existence of a contraction semigroup generated by the closure of $A$ on suitable $L^p$ spaces, $1\le p < \infty$ and on $C(\overline{\Omega}).$ Questions involving regularity are settled optimally, using De Giorgi-Nash-Moser iteration.

Article information

Adv. Differential Equations, Volume 11, Number 5 (2006), 481-510.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K60: Nonlinear initial value problems for linear parabolic equations
Secondary: 35K05: Heat equation 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]


Favini, Angelo; Ruiz Goldstein, Gisèle; Goldstein, Jerome A.; Romanelli, Silvia. The heat equation with nonlinear general Wentzell boundary condition. Adv. Differential Equations 11 (2006), no. 5, 481--510.

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