PORTUGALIAE MATHEMATICA Vol. 61, No. 4, pp. 375391 (2004) 

Energy decay estimates for the damped EulerBernoulli equation with an unbounded localizing coefficientL.R. Tcheugoué TébouDepartment of Mathematics, Florida International University,Miami FL 33199  USA Email: teboul@fiu.edu Abstract: We consider the EulerBernoulli equation in a bounded domain $\Omega$ with a local dissipation $ay'$. The localizing coefficient $a$ is of the form $a(x)={\alpha(x)/(d(x,\Gamma))^s}$, $(0<s\leq 1)$, where $\Gamma$ is the boundary of $\Omega$, $d(x,\Gamma)$ is the distance from $x$ to $\Gamma$, and $\alpha$ is a bounded nonnegative function such that $a$ is unbounded. Using integral inequalities and multiplier techniques, we prove exponential and polynomial decay estimates for the energy of each solution of this equation. In particular, since the localizing coefficient $a$ is unbounded, an important technical difficulty occurs adding to the difficulty of dealing with a local dissipation. A judicious application of Hardy inequality enables us to overcome this difficulty. The results obtained improve existing results where the boundedness of the function $a$ is critical. Keywords: EulerBernoulli equation; decay estimates; local dissipation; degenerate dissipation; integral inequalities; Hardy inequality; multiplier techniques. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2004 Sociedade Portuguesa de Matemática
