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

Abstract 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.