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Acta Mathematica Vietnamica

Existence and Asymptotic Behavior of Solutions for Degenerate Nonlinear Kirchhoff Strings with Variable-Exponent Nonlinearities

icon-email Rahmoune Abita

Abstract

In this paper, we investigate the existence of a local solution in time and discuss the exponential asymptotic behavior to a weakly damped wave equation involving the variable-exponents

uttM(|u(t)|2)Δu+0tg(ts)Δu(s)ds+γ1ut+|ut|k(x)1ut=|u|p(x)1u in Ω×R+

with simply supported boundary condition, where Ω is a bounded domain of Rn,g>0 is a memory kernel that decays exponentially, and M(s) is a locally Lipschitz function. This kind of problem without the memory term when k(.) and p(.) are constants models viscoelastic Kirchhoff equation.