Existence and nonexistence results for a class of pseudo-parabolic equations with combined logarithmic nonlinearities

In this paper, we study the existence and nonexistence results for a class of pseudo-parabolic equations with combined logarithmic nonlinearities γ | u | p- 2 u log | u | + u log | u |, where γ > 0 and p > 2 satisfy certain conditions. Employing the key ingredients in modified potential well method, we obtain the existence of a global weak solution and the finite time blow-up phenomenon for low initial energy and critical initial energy. We show that the global weak solution has exponential decay. An interesting observation is that the higher power term γ | u | p- 2 u log | u | ( p > 2) breaks the infinite time blow-up phenomenon of u log | u | with appropriate coefficient γ . Furthermore, an upper bound and a lower bound for the blow-up time are estimated. We give sufficient conditions for global existence and blow-up result of weak solutions in the case of high initial energy. Then we take account of the case when the initial energy is independent of the well depth. In addition, we show that the solution to the same pseudo-parabolic equation with only logarithmic nonlinearity u log | u | blows up at infinity.