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In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation u t t - u x x - a u x x t t + b u x 4 - d u x x t = f ( u ) x x are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.

This paper studies the existence and the non-existence of global solutions to the initial boundary value problems for the non-linear wave equation utt + uxxxx = σ(ux)x + f(x, t) and the Boussinesq-type equation utt + uxxxx = σ(u)xx + f(x, t). The paper proves that every above-mentioned problem has a unique global solution under rather mild confining conditions, and arrives at some sufficient conditions of blow-up of the solutions in finite time. Finally, a few examples are given. Copyright © 2000 John Wiley & Sons, Ltd.

The existence of global weak solutions to the periodic boundary problem or the initial value problem for the nonlinear Pseudo-hyperbolic equation $$u_{tt} - \left[ {a_1 + a_2 \left( {u_x } \right)^{2m} } \right]u_{xx} - a_3 u_{xxt} = f\left( {x,t,u,u_x } \right)$$ is proved by the method of the vanishing of the additional diffusion terms, Leray-Schauder's fixed-point argument and Sobolev's estimates, wherem≥1 is a natural number andai>0 (i=1, 2, 3) are constants.