GLOBAL EXISTENCE AND POINTWISE ESTIMATES OF SOLUTIONS FOR THE GENERALIZED SIXTH-ORDER BOUSSINESQ EQUATION

摘要

This paper studied the Cauchy problem for the generalized sixth-order Boussinesq equation in multi-dimension (n >= 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 | D(x)(alpha)u(x, t) | <= C(1 + t)(-n+|alpha|-1/2) (1 + |x|(2)/1+t)(-N) for N > [n/2] + 1.