Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density

Abstract

In this paper, we consider the initial boundary value problem of nonlinear evolution equation with hereditary memory, variable density, and external force term \t\t\t{|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.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} \\vert u_{t} \\vert ^{\\rho }u_{tt}-\\alpha \\Delta u-\\Delta u_{tt}+\\int_{-\\infty } ^{t}\\mu (t-s)\\Delta u(s)\\,ds-\\gamma \\Delta u_{t}=f(u), \\\\ \\quad (x,t)\\in \\varOmega \\times \\mathbb{R}^{+},\\\\ u(x,t)=0,\\quad (x,t)\\in \\partial \\varOmega \\times \\mathbb{R}^{+},\\\\ u(x,0)=u_{0}(x),\\qquad u_{t}(x,0)=u_{1}(x),\\quad x\\in \\varOmega. \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} 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.