Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 13 March 2017 / Approved: 13 March 2017 / Online: 13 March 2017 (09:56:13 CET)

A peer-reviewed article of this Preprint also exists.

Journal reference: Journal of Pseudo-Differential Operators and Applications 2017
DOI: 10.1007/s11868-017-0216-x

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 $I(u_{0}) \geq0$ 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.

Subject Areas

blow-up; semilinear pseudo-parabolic equations; critical initial energy; conical degeneration

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