The heat equation with nonlinear general Wentzell boundary condition

Abstract

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.