Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration

Abstract

In this paper, we study the semilinear pseudo-parabolic equations $$\displaystyle u_{t} - \triangle _{{\mathbb {B}}}u - \triangle _{{\mathbb {B}}}u_{t} = \left| u\right| ^{p-1}u$$ on a manifold with conical singularity, where $$\triangle _{{\mathbb {B}}}$$ 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 $$\displaystyle \Vert \nabla _{{\mathbb {B}}}u_{0}\Vert _{L_{2}^{\frac{n}{2}}({\mathbb {B}})} = 0$$ 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 $$I(u_{0}) \ge 0$$ 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.