Advances in Differential Equations

The heat equation with nonlinear general Wentzell boundary condition

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

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

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.

Export citation