Well-posedness and asymptotic behavior for a p-biharmonic pseudo-parabolic equation with logarithmic nonlinearity of the gradient type

This paper is concerned with the well-posedness and asymptotic behavior for an initial boundary value problem of a pseudo-parabolic equation with p-biharmonic operator and logarithmic nonlinearity of the gradient type. The existence of the global weak solution is established by combining the technique of potential-well and the method of Faedo-Galerkin approximation. Meantime, by virtue of the improved logarithmic Sobolev inequality and modified differential inequality, we obtain the results on infinite and finite time blow-up and derive the lifespan of blow-up solutions in various energy levels. Furthermore, the extinction phenomenon with extinction time is presented.