Cauchy problem for the multi-dimensional Boussinesq type equation

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt − u + 2u = σ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ ∈ C(R), σ(s) is of polynomial growth order, say p (> 1), either 0 σ(s)s β s 0 σ(τ)dτ, s ∈ R, whereβ >0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N = 1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown. © 2007 Elsevier Inc. All rights reserved