Well-posedness and Asymptotic Behavior for a Pseudo-parabolic Equation Involving $p$-biharmonic Operator and Logarithmic Nonlinearity

Abstract

This paper deals with the well-posedness and asymptotic behavior for a pseudo-parabolic equation involving $p$-biharmonic operator and logarithmic nonlinearity under Navior boundary condition. By combining Galerkin approximation, the method of potential well, the technique of differential inequality and improved logarithmic Sobolev inequality, we establish the local and global solvability, infinite and finite time blow-up phenomena of weak solutions in different energy levels. Moreover, we obtain the growth rate of weak solutions, life span in different energy cases and also give a result of extinction phenomenon.