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.