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:
[1]

M. BertschH. IzuharaM. Mimura and T. Wakasa, Standing and travelling waves in a parabolic-hyperbolic system, Discrete Contin. Dyn. Syst., 39 (2019), 5603-5635.  doi: 10.3934/dcds.2019246.  Google Scholar

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C. E. Kenig, The method of energy channels for nonlinear wave equations, Discrete Contin. Dyn. Syst., 39 (2019), 6979-6993.  doi: 10.3934/dcds.2019240.  Google Scholar

[6]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.   Google Scholar

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J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

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Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.   Google Scholar

[9]

M. Liu and C. Wang, Global existence for semilinear damped wave equations in relation with the Strauss conjecture, Discrete Contin. Dyn. Syst., 40 (2020), 709-724.   Google Scholar

[10]

Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.   Google Scholar

[11]

Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D, 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.  Google Scholar

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Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.  Google Scholar

[13]

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342.  doi: 10.1007/BF02572407.  Google Scholar

[14]

L. E. Payne and D. H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[15]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar

[16]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.  doi: 10.1006/jmaa.1999.6528.  Google Scholar

[17]

S. Wang and H. Xue, Global solution for a generalized Boussinesq equation, Appl. Math. Comput., 204 (2008), 130-136.  doi: 10.1016/j.amc.2008.06.059.  Google Scholar

[18]

R. Xu, Y. Chen, Y. Yang, S. Chen, J. Shen, T. Yu and Z. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations, Electron, J. Differential Equations, 2018 (2018), Paper No. 55, 52 pp.  Google Scholar

[19]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.  Google Scholar

[20]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[21]

R. Xu and Y. Yang, Global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations, Quart. Appl. Math., 71 (2013), 401-415.  doi: 10.1090/S0033-569X-2012-01295-6.  Google Scholar

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R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.   Google Scholar

show all references

References:
[1]

M. BertschH. IzuharaM. Mimura and T. Wakasa, Standing and travelling waves in a parabolic-hyperbolic system, Discrete Contin. Dyn. Syst., 39 (2019), 5603-5635.  doi: 10.3934/dcds.2019246.  Google Scholar

[2]

J. A. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 787-804.  doi: 10.3934/dcds.2004.10.787.  Google Scholar

[3]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[4]

R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27 (1996), 1165-1175.  doi: 10.1016/0362-546X(95)00119-G.  Google Scholar

[5]

C. E. Kenig, The method of energy channels for nonlinear wave equations, Discrete Contin. Dyn. Syst., 39 (2019), 6979-6993.  doi: 10.3934/dcds.2019240.  Google Scholar

[6]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.   Google Scholar

[7]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[8]

Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.   Google Scholar

[9]

M. Liu and C. Wang, Global existence for semilinear damped wave equations in relation with the Strauss conjecture, Discrete Contin. Dyn. Syst., 40 (2020), 709-724.   Google Scholar

[10]

Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.   Google Scholar

[11]

Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D, 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.  Google Scholar

[12]

Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.  Google Scholar

[13]

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342.  doi: 10.1007/BF02572407.  Google Scholar

[14]

L. E. Payne and D. H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[15]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar

[16]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.  doi: 10.1006/jmaa.1999.6528.  Google Scholar

[17]

S. Wang and H. Xue, Global solution for a generalized Boussinesq equation, Appl. Math. Comput., 204 (2008), 130-136.  doi: 10.1016/j.amc.2008.06.059.  Google Scholar

[18]

R. Xu, Y. Chen, Y. Yang, S. Chen, J. Shen, T. Yu and Z. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations, Electron, J. Differential Equations, 2018 (2018), Paper No. 55, 52 pp.  Google Scholar

[19]

R. XuW. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321-356.  doi: 10.1007/s11425-017-9280-x.  Google Scholar

[20]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[21]

R. Xu and Y. Yang, Global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations, Quart. Appl. Math., 71 (2013), 401-415.  doi: 10.1090/S0033-569X-2012-01295-6.  Google Scholar

[22]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.   Google Scholar

Figure 1.  $ d(\delta)\thicksim\delta $; $ p = 3 $, $ C_1 = 2 $
Figure 2.  $ d(0.5)\thicksim C_1, p $
Figure 3.  $ d(\delta)\thicksim\delta, p $; $ C_1 = 2 $
Figure 4.  $ g_1(y)\thicksim y_\delta $
Figure 5.  $ y_{1}\thicksim d(1), p $
Figure 6.  $ y_{0.5}\thicksim C_1, p $
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