Global Existence, Exponential Decay and Finite Time Blow-up of Solutions for a Class of Semilinear Pseudo-parabolic Equations with Conical Degeneration

In this paper, we study the semilinear pseudo-parabolic equations ut − 4Bu − 4But = |u|p−1 u on a manifold with conical singularity, where 4B is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary x1 = 0. Firstly, we 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 and nonexistence of global weak solution: for the low initial energy J(u0) < d, the solution is global in time with I(u0) > 0 or ‖∇Bu0‖ L n 2 2 (B) = 0 and blows up in finite time 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 in finite time with I(u0) < 0. The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.