Stability Estimates for Parabolic Problems with Wentzell boundary conditions

Of concern is the uniformly parabolic problem \begin{equation*} u_t =\dv(\A\nabla u),\qquad u(0,x)=f(x),\qquad u_t +\beta\pan u+\gamma u-q\beta \lb u=0, \end{equation*} for $x\in \Omega\subset \R^N$ and $t\ge0$. Here $\A=\{a_{ij}(x)\}_{ij}$ is a real, hermitian, uniformly positive definite $N\times N$ matrix; $\beta,\,\gamma\in C(\overline\Omega)$ with $\beta>0;\,q\in [0,\infty)$ and $\pan u$ is the conormal derivative of $u$ with respect to $A$: and everything is sufficiently regular. The solution of this well posed problem depends continuously on the ingredients of the problem, namely, $\A,\,\beta,\,\gamma,\,q,\, f.$ This is shown using semigroup methods in \cite{CFGGR}. More precisely, if we have a sequence of such problems with solutions $u_n$, and if $\A_n\to\A,\,\beta_n\to\beta,$ et al in a suitable sense, then $u_n\to u$, the solution of the limiting problem. The abstract analysis associated with operator semigroup theory gives this conclusion, but no rate of convergence. Determining how fast the convergence of the solutions is requires detailed estimates. Such estimates are provided in this paper.