Blow-Up of Solution to Cauchy Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

Changming Song,Li Chen

doi:10.4236/jamp.2015.37103

Abstract

We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.

Highlights

In [2], the author has discussed the nonexistence of global solution to the initial boundary value problem of Equation (1.1) in some condition
In order to prove that blow-up of Cauchy problem (1.1), (1.2), we shall consider the following auxiliary problem vtt = vxx + σ (vx )x + α vx4 + β vx6, x ∈ R, t > 0, (1.3)
It follows from lemma (2.1) that there exists t1

Summary

Introduction

We consider the following Cauchy problem utt= uxx + σ (u)xx + αux4 + β ux , x ∈ R, t > 0,. We can obtain blow-up of the Cauchy problem (1.1), (1.2) from (1.3), (1.4) by setting vx (x,t) = u(x,t) , How to cite this paper: Song, C.M. and Chen, L. F (v0x ) ∈ L1(R) , the solution v(x,t) of the auxiliary problem (1.3), (1.4) satisfies the following energy identity. The solution v(x,t) of the auxiliary problem (1.3), (1.4) blows-up in finite time if one of the following conditions holds (1) E(0) < 0;. It follows from lemma (2.1) that there exists. By virtue of assumption (2), we see H (0) > 0 and H (0) > 0 It follows from lemma (2.1) that there exists (3) If E(0) > 0 , by taking β0 = 0 , (2.9) becomes. H (t)H (t) − (1+ γ )H (t)2 ≥ −(2 + 4γ )E(0)H (t)

Defining then

Proof Let