Wellposedness of Nonlinear Parabolic Problems with nonlinear Wentzell boundary conditions

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