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In this paper, we study the semilinear pseudo-parabolic equations ut − B u − B ut = |u| p−1 u on a manifold with conical singularity, where B is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary x 1 = 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(u 0 ) < d, the solution is global in time with I(u 0 ) > 0 or ∇ B u 0= 0 and blows up in finite time with I(u 0 ) < 0; for the critical initial energy J(u 0 ) = d, the solution is global in time with I(u 0 ) ≥ 0 and blows up in finite time with I(u 0 ) < 0. The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.2010 Mathematics Subject Classification. 26D15, 41A55. Key words and phrases. blow-up; semilinear pseudo-parabolic equations; critical initial energy; conical degeneration.

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