LAUSR

AbstractIn this paper, we consider the initial boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force term {|ut|ρutt−αΔu−Δutt+∫−∞tμ(t−s)Δu(s)ds−γΔut=f(u),(x,t)∈Ω×R+,u(x,t)=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω. $$\begin{aligned} \textstyle\begin{cases} \vert u_{t} \vert ^{\rho }u_{tt}-\alpha \Delta u-\Delta u_{tt}+\int_{-\infty } ^{t}\mu (t-s)\Delta u(s)\,ds-\gamma \Delta u_{t}=f(u), \\ \quad (x,t)\in \varOmega \times \mathbb{R}^{+},\\ u(x,t)=0,\quad (x,t)\in \partial \varOmega \times \mathbb{R}^{+},\\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\quad x\in \varOmega. \end{cases}\displaystyle \end{aligned}$$ Under suitable assumptions, we prove the existence of a global solution by means of the Galerkin method, establish the exponential stability result by using only one simple auxiliary functional, and give the polynomial stability result.