Abstract:

In this paper, we prove that the Cauchy problem for a generalized system of the coupled Boussinesq-type equations arising from nonlinear layered lattice model $\begin{eqnarray} &u_{tt}-a_{1}u_{xx}-a_{2}u_{xxtt}+a_{3}(u-w)=f(u_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0, \nonumber\ &w_{tt}-b_{1}w_{xx}-b_{2}w_{xxtt}+b_{3}(w-u)=g(w_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0\nonumber \end{eqnarray} $ has a unique global generalized solution in $C([0, \infty);H^s(\ {\Bbb R})\times H^s(\ {\Bbb R}))(s\geq2$ is a real number) and a unique global classical solution in $C^2([0, \infty);C_B^2(\ {\Bbb R})\times C_B^2(\ {\Bbb R}))(s>\frac{5}{2})$. The sufficient conditions for the blow up of the solution to the Cauchy problem above are given.

Key words: Generalized system of coupled Boussinesq-type equations, Cauchy problem, Global solution, Blow up of solution