General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions

Mi Jin Lee and Jum-Ran Kang ()
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Mi Jin Lee: Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea
Jum-Ran Kang: Department of Applied Mathematics, Pukyong National University, Busan 48513, Republic of Korea

Mathematics, 2023, vol. 11, issue 22, 1-21

Abstract: This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition a 2 > 0 and without imposing any restrictive growth assumption on the damping term f 1 , using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function ψ , namely ψ ′ ( t ) ≤ − μ ( t ) G ( ψ ( t ) ) , where G is a convex and increasing function near the origin, and μ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions μ and G , as well as the function F defined by f 0 , which characterizes the growth behavior of f 1 at the origin.

Keywords: optimal decay; viscoelastic wave equation; nonlinear time-varying delay; nonlinear damping; acoustic boundary conditions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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