Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient (Q1769301)

The paper deals with the problem \(y''+\Delta^2 y+a y' = 0\) in \(\Omega\times(0,\infty)\), \(y=\frac{\partial y}{\partial\nu}=0\) on \(\Gamma\times(0,\infty)\), \(y(0)=y^0\) in \(\Omega\), \(y'(0)=y^1\) in \(\Omega\), where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with sufficiently smooth boundary \(\Gamma\), \(a(x)\) is a nonnegative function that blows up on a certain part of \(\Gamma\). The author first reviews a known existence result and then derive decay estimates for the energy \(E(t)=\frac{1}{2}\int_\Omega (| y'| ^2 + | \Delta y| ^2)\,dx\) of a solution \(y(x,t)\). He considers two different cases based on the properties of \(a(x)\) and the regularity of initial conditions. In the proof the author tackles various technical issues, among the used tools is the Hardy inequality and a solution of an auxiliary problem.