Weighted integral inequality and applications in general energy decay estimate for a variable density wave equation with memory

Abstract This paper develops a weighted integral inequality to derive decay estimates for the quasilinear viscoelastic wave equation with variable density |ut|ρutt−Δu−Δutt+∫0tg(t−s)Δu(s)ds=0in Ω×(0,∞) $$\begin{aligned} \vert u_{t} \vert ^{\rho }u_{tt}-\Delta u-\Delta u_{tt}+ \int^{t}_{0}g(t-s)\Delta u(s)\,ds=0 \quad \text{in } \varOmega \times (0, \infty ) \end{aligned}$$ with initial conditions and boundary condition, where g is a memory kernel function and ρ is a positive constant. Depending on the properties of convolution kernel g at infinity, we establish a general decay rate of the solution such that the exponential and polynomial decay results in some literature are special cases of this paper, and we improve the integral method used in the literature.