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May  2019, 18(3): 1351-1358. doi: 10.3934/cpaa.2019065

Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

Yanbing Yang 1,2,3, and  Runzhang Xu 1,,

1. 

College of Science, Harbin Engineering University, 150001, China
2. 

College of Computer Science and Technology, Harbin Engineering University, 150001, China
3. 

Department of Mathematics, University of Texas, Arlington, TX 76019, USA

* Corresponding author

Received  June 2018 Revised  September 2018 Published  November 2018

Fund Project: The first author was supported by the National Natural Science Foundation of China (11801114), the Heilongjiang Postdoctoral Foundation (LBH-Z15036), the China Scholarship Council (201706685064), the Fundamental Research Funds for the Central Universities. The second author was supported by the National Natural Science Foundation of China (11871017), the China Postdoctoral Science Foundation (2013M540270), the Fundamental Research Funds for the Central Universities.

By introducing a new increasing auxiliary function and employing the adapted concavity method, this paper presents a finite time blow up result of the solution for the initial boundary value problem of a class of nonlinear wave equations with both strongly and weakly damped terms at supercritical initial energy level.

Keywords: Wave equation, initial boundary value problem, strong damping term, finite time blow up, supercritical initial energy.
Mathematics Subject Classification: Primary: 35L05, 35L20, 35B44.
Citation: Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
References:
[1]

B. Bilgin and V. Kalantarov, Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94.  doi: 10.1016/j.jmaa.2013.01.056.  Google Scholar

[2]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar

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H. Chen and G. Liu, Well-posedness for a class of Kirchhoff equations with damping and memory terms, IMA J. Appl. Math., 80 (2015), 1808-1836.  doi: 10.1093/imamat/hxv018.  Google Scholar

[4]

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

[5]

S. Gerbi and B. Said-Houari, Exponential decay for solutions to semilinear damped wave equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 559-566.  doi: 10.3934/dcdss.2012.5.559.  Google Scholar

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S. Gerbi and B. Said-Houari, Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.   Google Scholar

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P. Graber and B. Said-Houari, Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122.  doi: 10.1007/s00245-012-9165-1.  Google Scholar

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G. Liu and S. Xia, Global existence and finite time blow up for a class of semilinear wave equations on R-N, Comput. Math. Appl., 70 (2015), 1345-1356.  doi: 10.1016/j.camwa.2015.07.021.  Google Scholar

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Y. Liu and R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar

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G. Philippin, Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., 66 (2015), 129-134.  doi: 10.1007/s00033-014-0400-2.  Google Scholar

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B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (2010), 79-92.   Google Scholar

[12]

J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043. doi: 10.1142/S0129167X13500432.  Google Scholar

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H. Song and D. Xue, Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 19 (2014), 245-251.  doi: 10.1016/j.na.2014.06.012.  Google Scholar

[14]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015.  Google Scholar

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Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.  doi: 10.1090/S0002-9939-08-09514-2.  Google Scholar

[16]

Y. Wang, Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA J. Appl. Math., 74 (2009), 392-415.  doi: 10.1093/imamat/hxp004.  Google Scholar

[17]

T. Wick, Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity, Comput. Mech., 53 (2014), 29-43.  doi: 10.1007/s00466-013-0890-3.  Google Scholar

[18]

R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.   Google Scholar

[19]

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

[20]

R. Xu, Y. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.  Google Scholar

[21]

R. Xu, Y. Yang, B. Liu, J. Shen and S. Huang, Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015), 955-976.  doi: 10.1007/s00033-014-0459-9.  Google Scholar

[22]

X. Zhu, F. Li and T. Rong, Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun. Pure Appl. Anal., 14 (2015), 2465-2485.  doi: 10.3934/cpaa.2015.14.2465.  Google Scholar

show all references

References:
[1]

B. Bilgin and V. Kalantarov, Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations, J. Math. Anal. Appl., 403 (2013), 89-94.  doi: 10.1016/j.jmaa.2013.01.056.  Google Scholar

[2]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar

[3]

H. Chen and G. Liu, Well-posedness for a class of Kirchhoff equations with damping and memory terms, IMA J. Appl. Math., 80 (2015), 1808-1836.  doi: 10.1093/imamat/hxv018.  Google Scholar

[4]

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

[5]

S. Gerbi and B. Said-Houari, Exponential decay for solutions to semilinear damped wave equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 559-566.  doi: 10.3934/dcdss.2012.5.559.  Google Scholar

[6]

S. Gerbi and B. Said-Houari, Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.   Google Scholar

[7]

P. Graber and B. Said-Houari, Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122.  doi: 10.1007/s00245-012-9165-1.  Google Scholar

[8]

G. Liu and S. Xia, Global existence and finite time blow up for a class of semilinear wave equations on R-N, Comput. Math. Appl., 70 (2015), 1345-1356.  doi: 10.1016/j.camwa.2015.07.021.  Google Scholar

[9]

Y. Liu and R. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations, 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.  Google Scholar

[10]

G. Philippin, Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., 66 (2015), 129-134.  doi: 10.1007/s00033-014-0400-2.  Google Scholar

[11]

B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (2010), 79-92.   Google Scholar

[12]

J. Shen, Y. Yang, S. Chen and R. Xu, Finite time blow up of fourth-order wave equations with nonlinear strain and source terms at high energy level, Internat. J. Math., 24 (2013), 1350043. doi: 10.1142/S0129167X13500432.  Google Scholar

[13]

H. Song and D. Xue, Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 19 (2014), 245-251.  doi: 10.1016/j.na.2014.06.012.  Google Scholar

[14]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015.  Google Scholar

[15]

Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.  doi: 10.1090/S0002-9939-08-09514-2.  Google Scholar

[16]

Y. Wang, Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA J. Appl. Math., 74 (2009), 392-415.  doi: 10.1093/imamat/hxp004.  Google Scholar

[17]

T. Wick, Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity, Comput. Mech., 53 (2014), 29-43.  doi: 10.1007/s00466-013-0890-3.  Google Scholar

[18]

R. Xu, Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data, Quart. Appl. Math., 68 (2010), 459-468.   Google Scholar

[19]

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

[20]

R. Xu, Y. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.  Google Scholar

[21]

R. Xu, Y. Yang, B. Liu, J. Shen and S. Huang, Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015), 955-976.  doi: 10.1007/s00033-014-0459-9.  Google Scholar

[22]

X. Zhu, F. Li and T. Rong, Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source, Commun. Pure Appl. Anal., 14 (2015), 2465-2485.  doi: 10.3934/cpaa.2015.14.2465.  Google Scholar

Table 1.  Obtained results and open problems for problem (1)-(3)
Global existence Asymptotic behavior Blow up
Subcritical initial energy $E(0)<d$ Reference [4] Reference [4] Reference [4]
Critical initial energy $E(0)=d$ Reference [4] Reference [4] Reference [4]
Supercritical initial energy $E(0)>d$ Open problem Open problem $\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
Reference [4]		 $\omega>0$, $\mu>0$
$\omega>0$, $\mu=0$
$\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
Present paper
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Global existence Asymptotic behavior Blow up
Subcritical initial energy $E(0)<d$ Reference [4] Reference [4] Reference [4]
Critical initial energy $E(0)=d$ Reference [4] Reference [4] Reference [4]
Supercritical initial energy $E(0)>d$ Open problem Open problem $\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
Reference [4]		 $\omega>0$, $\mu>0$
$\omega>0$, $\mu=0$
$\omega=0$, $\mu>0$
$\omega=0$, $\mu=0$
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