Stability of Parabolic Problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear uniformly parabolic problem \begin{equation*} u_t =\dv(\A\nabla u),\qquad u(0,x)=f(x),\qquad u_t +\beta\pan u+\gamma(x, u)-q\beta \lb u=0, \end{equation*} 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; $\beta\in C(\p\Omega)$ with $\beta>0$; $\gamma:\p\Omega\times\R\to \R; \,q\in [0,\infty)$ and $\pan u$ is the conormal derivative of $u$ with respect to $A$: and everything is sufficiently regular. The solution of this wellposed problem depends continuously on the ingredients of the problem, namely, $\A,\,\beta,\,\gamma,\,q,\, f.$ This is shown using semigroup methods in \cite{CFGGR}. Here we prove explicit stability estimates of the solution $u$ with respect to the coefficients $\A,\,\beta,\,\gamma,\,q,$ and the initial condition $f$. Moreover we cover the singular case of a problem with $q=0$ which is approximated by problems with positive $q$.