Global existence and exponential growth of solution for the logarithmic Boussinesq-type equation

In this paper we consider the Boussinesq-type equation associated with logarithmic nonlinear term and initial boundary value conditions. By using potential well method combined with the logarithmic Sobolev inequality, we obtain the existence of global solution. At last we show that the L2L2-norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions for initial data. The result shows that the polynomial nonlinearity is a critical condition of blow-up in finite time for the solutions of Boussinesq equations.