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

L.R. Tcheugoué Tébou

Abstract: We consider the Euler--Bernoulli 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: Euler--Bernoulli equation; decay estimates; local dissipation; degenerate dissipation; integral inequalities; Hardy inequality; multiplier techniques.

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