On decay properties of solutions for degenerate Kirchhoff equations with strong damping and source terms

Shun-Tang Wu

doi:10.1186/1687-2770-2012-93

Abstract

We investigate the degenerate Kirchhoff equations with strong damping and source terms of the form in a bounded domain. We obtain the optimal decay rate for by deriving its decay estimate from below, provided that either ε is suitably small or the initial data satisfy the proper smallness assumption. The key ingredient in the proof is based on the work of Ono (J. Math. Anal. Appl. 381(1):229-239, 2011), with necessary modification imposed by our problem. MSC:35L70, 35L80.

Highlights

In this paper, we consider the initial boundary value problem for the following degenerate Kirchhoff equations with strong damping and source terms: εutt – γ u– ut = f (u), in × (, ∞), ( . )u(x, ) = u (x), ut(x, ) = u (x), x ∈, u(x, t) =, x ∈ ∂, t ≥, where is a bounded domain in RN (N ≥ ) with a smooth boundary ∂
In the case N =, the nonlinear vibrations of the elastic string are written in the form:
They proved the global existence of a unique solution under the small data condition in H ( ) ∩ H ( ) × H ( )

Summary

Introduction

We consider the initial boundary value problem for the following degenerate Kirchhoff equations with strong damping and source terms: εutt – γ u–. Hosoya and Yamada [ ] studied the following equation: utt – M u + δut = , with M(r) ≥ M > (the nondegenerate case) They proved the global existence of a unique solution under the small data condition in H ( ) ∩ H ( ) × H ( ). ). Later, Nishihara [ ] established a decay estimate from below of the potential of solutions to problem Motivated by these works, in this paper, we intend to give the optimal decay estimate for ∇ut to problem Let u satisfy the assumptions of Lemma.

If ω

Applying with ω