Nonlinear elliptic systems with dynamic boundary conditions (Q1204275)

We investigate strongly coupled semilinear elliptic systems on a smooth and bounded domain \(\Omega\) in \(\mathbb{R}^ n\) with nonlinear dynamic boundary conditions of the following form: \[ (P_ \lambda)\qquad \begin{cases}\lambda u-\partial_ j(a_{jk}\partial_ k u)+a_ j \partial_ j u+a_ 0 u=f(u,\nabla u) &\text{in } \Omega\times(0,\infty),\\ \partial_ t u+a_{jk}\nu^ j \partial_ k u+bu=g(u) &\text{on } \partial\Omega\times(0,\infty),\\ u(\cdot,0)=z_ 0 &\text{on }\partial\Omega.\end{cases} \] We show that there is a \(\lambda_ 0\geq 0\) such that for each \(\lambda\geq\lambda_ 0\) problem \((P_ \lambda)\) possesses a unique local solution, provided \(f\) and \(g\) satisfy weak regularity assumptions. Furthermore we specify conditions (essentially appropriate growth rates) on \(f\) and \(g\) which imply global existence in time of the corresponding solution. From a functional analytical point of view these results are obtained by using the theory of parabolic evolution operators and the theory of linear interpolation.