Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we study the fractional pseudo-parabolic equations ut + (−△)s u + (−△)s ut = u log |u|. Firstly, we recall the relationship between the fractional Laplace operator (−△)s and the fractional Sobolev space Hs and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of global weak solution: for the low initial energy J(u0) < d, the solution is global in time with I(u0) > 0 or ∥u0∥X0(Ω) = 0 and blows up +∞ with I(u0) < 0; for the critical initial energy J(u0) = d, the solution is global in time with I(u0) ≥ 0 and blows up at +∞ with I(u0) < 0. The decay estimate of the energy functional for the global solution is also given.