Asymptotic behavior of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms

Tengyu Ma,Longsuo Li,Juan Gu

doi:10.1186/s13660-016-1269-8

Abstract

We study the long time asymptotic behavior of solutions to a class of fourth-order nonlinear evolution equations with dispersive and dissipative terms. By using the integral estimation method combined with the Gronwall inequality, we point out that the global strong solutions of the problems decay to zero exponentially with the passage of time to infinity. The proof is rigorous and only based on some relatively weak assumptions on the nonlinear term.

Highlights

Nonlinear pseudo-hyperbolic equations were proposed from biological and mechanical problems in recent years, such as nerve conduction and the longitudinal vibration of rods with viscous effects, which have important practical and theoretical backgrounds.More and more attention is paid to nonlinear evolution equations with dispersive and dissipative terms because these problems arise widely in practical applications
In, Zhu [ ] investigated the communication of longitudinal deformation wave for a flexible rod, he considered the influence of fourth-order nonlinear evolution equations with nonlinear, dispersive and dissipative terms, utt – C + nanunx– uxx – βuxxtt = γ uxxt, where C, γ >, β >, and an = are constants
In [, ] Saxton and Hrusa discussed the well-posedness of local solution and the existence of a global solution

Summary

Introduction

Nonlinear pseudo-hyperbolic equations were proposed from biological and mechanical problems in recent years, such as nerve conduction and the longitudinal vibration of rods with viscous effects, which have important practical and theoretical backgrounds.More and more attention is paid to nonlinear evolution equations with dispersive and dissipative terms because these problems arise widely in practical applications. In , Zhu [ ] investigated the communication of longitudinal deformation wave for a flexible rod, he considered the influence of fourth-order nonlinear evolution equations with nonlinear, dispersive and dissipative terms, utt – C + nanunx– uxx – βuxxtt = γ uxxt, where C , γ > , β > , and an = are constants. In [ ] Liu and Zhao considered the global existence of W k,p solutions to wave equations with a dispersive term.

Results

Conclusion