General decay rates for the wave equation with mixed-type damping mechanisms on unbounded domain with finite measure

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 66; no. 6; pp. 3123 - 3145
• Main Authors: Dias Silva, Flávio R., Nascimento, Flávio A. F., Rodrigues, José H.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2015
• Subjects: Damping; Decay rate; Engineering; Estimates; Functions (mathematics); Mathematical analysis; Mathematical Methods in Physics; Nonlinearity; Polynomials; Theoretical and Applied Mechanics; Wave equations; Wave equation; Unbounded domain; Stability; 35L70; 35B35; Frictional damping; 74D10; Viscoelastic damping
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-015-0547-5

This paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation subject to a locally distributed viscoelastic dissipation and a nonlinear frictional damping u t t - Δ u + ∫ 0 t g ( t - s ) div [ a ( x ) ∇ u ( s ) ] d s + b ( x ) f ( u t ) = 0 on Ω × ] 0 , ∞ [ , where Ω ⊂ R n , n ≥ 2 is an unbounded open set with finite measure and unbounded smooth boundary ∂ Ω = Γ . Supposing that the localization functions satisfy the “competitive” assumption a ( x ) + b ( x ) ≥ δ > 0 for all x ∈ Ω and the relaxation function g satisfies certain nonlinear differential inequalities introduced by Lasiecka et al. (J Math Phys 54(3):031504, 2013 ), we extend to our considered domain the prior results of Cavalcanti and Oquendo (SIAM J Control Optim 42(4):1310–1324, 2003 ). In addition, while in Cavalcanti and Oquendo ( 2003 ) the authors just consider exponential and polynomial decay rate estimates, in the present article general decay rate estimates are obtained.