Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term

Nadia Mezouar,Salah Boulaaras

doi:10.1186/s13661-020-01390-9

Abstract

The paper studies a system of nonlinear viscoelastic Kirchhoff system with a time varying delay and general coupling terms. We prove the global existence of solutions in a bounded domain using the energy and Faedo–Galerkin methods with respect to the condition on the parameters in the coupling terms together with the weight condition as regards the delay terms in the feedback and the delay speed. Furthermore, we construct some convex function properties, and we prove the uniform stability estimate.

Highlights

The Kirchhoff equation belongs to the famous wave equation’s models describing the transverse vibration of a string fixed in its ends
Where the function u(x, t) is the vertical displacement at the space coordinate x, varying in the segment [0, L] and over time t > 0, ρ is the mass density, h is the area of the cross section of the string, P0 is the initial tension on the string, L is the length of the string and E is the Young modulus of the material
When we do not have an initial tension (i.e. P0 = 0), we call that a degenerate case as opposed to the non-degenerate case

Summary

Introduction

The Kirchhoff equation belongs to the famous wave equation’s models describing the transverse vibration of a string fixed in its ends. A lot of work has been published with this term, for example see [11] and [14], where we find different results about the global existence and nonexistence of solutions and the decay of energy. Under the assumptions set on g1, g2, σ and τ , the authors have gotten the global existence of a solution and the decay rate of the energy.

Results

Conclusion