On a semilinear pseudo-parabolic equation with nonlinear convolution terms

This paper deals with the well-posedness and blow-up phenomena for a semilinear pseudo- parabolic equation with a nonlinear convolution term under the null Dirichlet boundary condition. By Hardy-Littlewood-Sobolev inequality, together with contraction mapping principle and the family of potential wells, we establish the local solvability and obtain the threshold between the existence and nonexistence of the global solution with low initial energy. Meantime, based on the modified differential inequality technique, the results of blow-up with arbitrary initial energy and the upper bound of lifespan are presented.