A uniform coerciveness result for biharmonic operator and its application to a parabolic equation (Q2803010)

Under consideration is the problem NEWLINE\[NEWLINE \Delta^{2} u=f(x),\quad \frac{\partial u}{\partial n}|_{\partial \Omega}=0, \quad -\frac{\partial \Delta u}{\partial n}+ \alpha\beta u|_{\partial \Omega}=0, \eqno{(1)} NEWLINE\]NEWLINE where \(n\) is the outward unit normal vector to \(\partial\Omega\), \(\alpha\) is a positive function, \(\beta\) is a nonnegative parameter, and \(\Omega\) is a bounded domain with smooth boundary. The author establishs an a priori estimate \(\|u\|_{W_{2}^{4}(\Omega)}\leq K(\|u\|_{L_{2}(\Omega)}+ \|f\|_{L_{2}(\Omega)})\) with a constant \(K\) independent of \(\beta\) which allows to pass to the limit as \(\beta\to\infty\). The results are applied to the study of the large-time behavior of a solution to the equation \(u_{t}+\Delta^{2}u = f(x, t)\) in an asymptotically cylindrical domain \(D\) satisfying the boundary conditions those in (1) with \(\beta=\beta(t)\to \infty\) as \(t\to \infty\).