Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions

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 $q$.