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

Abstract

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 $$u_{tt}- \Delta u+ \int_0^t g(t-s){\rm div}[a(x)\nabla u(s)]\,{\rm d}s + b(x) f(u_t)=0\,\quad {\rm on} \quad \Omega\times]0,\infty[,$$ where \({\Omega\subset\mathbb{R}^n, n\geq 2}\) is an unbounded open set with finite measure and unbounded smooth boundary \({\partial\Omega = \Gamma}\). Supposing that the localization functions satisfy the “competitive” assumption \({a(x)+b(x)\geq\delta>0}\) for all \({x\in \Omega}\) 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.