Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

Abstract

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.

Global Attractors for the Higher-Order Evolution Equation

“…Lately, Xie and Zhong in [8] studied the existence of global attractor of solution for the problem (1.1) with f = 0. Also, there are some authors studied the existence and nonexistence, asymptotic behavior of global solution for (1.2) (see [2][3][4][5][6][7] for more details ). Nakao and Yang in [9] showed the global attractor of the Kirchhoff type wave equation.…”

Superconvergence of a two‐grid method for fourth‐order dispersive‐dissipative wave equations

“…On one hand, the fourth‐order wave Equation demonstrates the spread problems of longitudinal strain waves in the nonlinear elastic rods and the ion acoustic of space transformation by weak nonlinear effect. () There have been some studies of theoretical analysis() and numerical simulations() devoted to this problem. More precisely, proved the existence and uniqueness of global strong solution under certain conditions that f ∈ C 1 , f ′ ( u ) is bounded above and satisfies | f ′ ( u )| ≤ A | u | p + B ,0< p < ∞ , A , B are constants.…”

Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms

“…As introduced in Ref. [25], some nonlinear evolution as the main form u tt − u xx − u xxtt and different nonlinear terms were obtained when we studied the spread of longitudinal strain waves in the nonlinear elastic rods and the weakly nonlinear ion acoustic and space-charge waves. In Ref.…”

“…This result has been extended by Liu and Li [13] to the case of all n 1. Recently, Xu et al [25] carried out a multiplier method and proved that the global strong solution of problem (1.4) decays to zero exponentially as t → +∞.…”

“…Our methods to prove these results are motivated by Refs. [3,5,16,25] while we should overcome the additional difficulties brought by the treatment of the nonlinear coupled terms and the canonical forms (ρ > 0). This paper is organized as follows.…”