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.