Communications in Mathematical Sciences

Volume 15 (2017)

Number 5

Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation

Pages: 1457 – 1487

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n5.a11

Authors

Changhong Guo (School of Management, Guangdong University of Technology, Guangzhou, China)

Shaomei Fang (Department of Mathematics, South China Agricultural University, Guangzhou, China)

Abstract

This paper studied the Cauchy problem for the generalized sixth-order Boussinesq equation in multi-dimension ($n \geq 3$), which was derived in the shallow fluid layers and nonlinear atomic chains. Firstly the global classical solution for the problem is obtained by means of long wave-short wave decomposition, energy method and the Green’s function. Secondly and what’s more, the pointwise estimates of the solutions are derived by virtue of the Fourier analysis and Green’s function, which concludes that $\lvert D^{\alpha}_x \: u(x,t) \rvert \leq C(1+t)^{-\dfrac{n + \lvert \alpha \rvert - 1}{2}} {\left ( 1+ \dfrac{{\lvert x \rvert}^2}{1+t} \right)}^{-N}$ for $N \gt \Bigl [ \dfrac{n}{2} \Bigr ] +1$.

Keywords

global existence, pointwise estimates, generalized sixth-order Boussinesq equation, Green’s function

2010 Mathematics Subject Classification

35B40, 35G25, 35Q35

Received 27 October 2016

Published 26 June 2017