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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 L 2 -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. [ABSTRACT FROM AUTHOR]

In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation u t t − Δ u − Δ u t t + Δ 2 u + Δ f ( u ) = 0 . Under the assumption that f ( u ) is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence and nonexistence of global weak solution. There are very few works on Boussinesq equation with nonlinear exponential growth term by potential well theory. [ABSTRACT FROM AUTHOR]