Cauchy problem for the multi-dimensional Boussinesq type equation

Abstract: The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation . It proves that the Cauchy problem admits a global weak solution under the assumptions that , is of polynomial growth order, say p (>1), either , , where 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 . In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown. [Copyright &y& Elsevier]