Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result

By Marcelo Moreira Cavalcanti, Valéria N. Domingos Cavalcanti, Ryuichi Fukuoka, J. A. Soriano. 2008

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This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by u tt - Δ M u + α(x) g(u t ) = 0 on M × ]0, ∞[, where M C R 3 is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced on M by R 3 , we prove that for each ∈ > 0, there exist an open subset V C M and a smooth function f: M→ R such that meas(V) ≥ meas(M) - e, Hessf ≈ g on V and inf Δf(x)| > 0. x∈v In addition, we prove that if α(x) ≥ a o > 0 on an open subset M* C M which contains M\V and if g is a monotonic increasing function such that k|s| ≤ g(s)| ≤ K|s| for all |s| > 1, then uniform and optimal decay rates of the energy hold.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd47699dfd58f4f2d585968f2bbf9feb9c4abd30