# American Institute of Mathematical Sciences

June  2020, 28(2): 807-820. doi: 10.3934/era.2020041

## A family of potential wells for a wave equation

 1 College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730124, China 2 College of Power and Energy Engineering, Harbin Engineering University, Harbin, Heilongjiang 150001, China

* Corresponding author: Wenke Li, liwenke@hrbeu.edu.cn

Received  March 2020 Revised  April 2020 Published  May 2020

In this paper, a family of potential wells are introduced by means of the modified depths of the potential wells. These potential wells are employed to study the initial-boundary value problem for a wave equation. The expression of the depths of the potential wells is derived. Global existence and finite time blow-up of weak solutions with the subcritical initial energy and the critical initial energy are obtained, respectively. Moreover, some numerical simulations of the depths of the potential wells are carried out.

Citation: Yang Liu, Wenke Li. A family of potential wells for a wave equation. Electronic Research Archive, 2020, 28 (2) : 807-820. doi: 10.3934/era.2020041
##### References:

show all references

##### References:
$d(\delta)\thicksim\delta$; $p = 3$, $C_1 = 2$
$d(0.5)\thicksim C_1, p$
$d(\delta)\thicksim\delta, p$; $C_1 = 2$
$g_1(y)\thicksim y_\delta$
$y_{1}\thicksim d(1), p$
$y_{0.5}\thicksim C_1, p$
 [1] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [2] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [3] Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077 [4] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [5] Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051 [6] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [7] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [8] Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 [9] Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 [10] Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 [11] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [12] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [13] Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535 [14] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [15] Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043 [16] Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169 [17] Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089 [18] Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351 [19] Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105 [20] Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143

2018 Impact Factor: 0.263