Electron. J. Differential Equations, Vol. 2018 (2018), No. 18, pp. 1-27.

Asymptotic stability and blow-up of solutions for an edge-degenerate wave equation with singular potentials and several nonlinear source terms of different sign

Feida Jiang, Yue Luan, Gang Li

Abstract:
We study the initial boundary value problem of an edge-degenerate wave equation. The operator $\Delta_{\mathbb{E}}$ with edge degeneracy on the boundary $\partial E$ was investigated in the literature. We give the invariant sets and the vacuum isolating behavior of solutions by introducing a family of potential wells. We prove that the solution is global in time and exponentially decays when the initial energy satisfies $E(0)\leq d$ and $I(u_0)>0$. Moreover, we obtain the result of blow-up with initial energy $E(0)\leq d$ and $I(u_0)<0$, and give a lower bound for the blow-up time $T^*$.

Submitted May 12, 2017. Published January 13, 2018.
Math Subject Classifications: 35L20, 35L80, 33B44.
Key Words: Edge-degenerate; potential function; blow-up; decay; global solution.

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Feida Jiang
College of Mathematics and Statistics
Nanjing University of Information Science and Technology
Nanjing 210044, China
email: jfd2001@163.com
Yue Luan
College of Mathematics and Statistics
Nanjing University of Information Science and Technology
Nanjing 210044, China
email: yluan_sunflower@163.com
Gang Li
College of Mathematics and Statistics
Nanjing University of Information Science and Technology
Nanjing 210044, China
email: ligang@nuist.edu.cn

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