doi: 10.3934/eect.2021019

Initial boundary value problem for a strongly damped wave equation with a general nonlinearity

School of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Yuzhu Han

Received  October 2020 Revised  February 2021 Early access  April 2021

Fund Project: The first author is supported by NSFC grant 11401252 and by The Education Department of Jilin Province grant JJKH20190018KJ

In this paper, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.

Citation: Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, doi: 10.3934/eect.2021019
References:
[1]

E. BelchevM. Kepka and Z. F. Zhou, Finite-time blow-up of solutions to semilinear wave equations, J. Funct. Anal., 190 (2002), 233-254.  doi: 10.1006/jfan.2001.3885.  Google Scholar

[2]

H. F. Di, Y. D. Shang and Z. F. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl., 51 (2020), 102968, 22pp. doi: 10.1016/j.nonrwa.2019.102968.  Google Scholar

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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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Y. Z. HanW. J. GaoZ. Sun and H. X. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar

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H. A. Levine, Remarks on the growth and nonexistence of solutions to nonlinear wave equations, A Seminar on PDEs-1973, Rutgers Univ., New Brunswick, N. J., 1973, 59–70.  Google Scholar

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K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0.  Google Scholar

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V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[11]

T. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4.  Google Scholar

[12]

L. L. SunB. Guo and W. J. Gao, A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., 37 (2014), 22-25.  doi: 10.1016/j.aml.2014.05.009.  Google Scholar

[13]

G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643.  doi: 10.4153/CJM-1980-049-5.  Google Scholar

[14]

Y. B. Yang and R. Z. Xu, Nonlinear wave equation with both strong and weakly damped terms: supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065.  Google Scholar

[15]

J. Zhou, Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett., 45 (2015), 64-68.  doi: 10.1016/j.aml.2015.01.010.  Google Scholar

[16]

G. Zu and B. Guo, Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy, Evol. Equ. Control The., 10 (2021), 259-270.  doi: 10.3934/eect.2020065.  Google Scholar

show all references

References:
[1]

E. BelchevM. Kepka and Z. F. Zhou, Finite-time blow-up of solutions to semilinear wave equations, J. Funct. Anal., 190 (2002), 233-254.  doi: 10.1006/jfan.2001.3885.  Google Scholar

[2]

H. F. Di, Y. D. Shang and Z. F. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl., 51 (2020), 102968, 22pp. doi: 10.1016/j.nonrwa.2019.102968.  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]

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

[5]

B. Guo and F. Liu, A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett., 60 (2016), 115-119.  doi: 10.1016/j.aml.2016.03.017.  Google Scholar

[6]

Y. Z. HanW. J. GaoZ. Sun and H. X. Li, Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy, Comput. Math. Appl., 76 (2018), 2477-2483.  doi: 10.1016/j.camwa.2018.08.043.  Google Scholar

[7]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equation of the form $Pu_t = -Au+\mathcal{F}(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.  Google Scholar

[8]

H. A. Levine, Remarks on the growth and nonexistence of solutions to nonlinear wave equations, A Seminar on PDEs-1973, Rutgers Univ., New Brunswick, N. J., 1973, 59–70.  Google Scholar

[9]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0.  Google Scholar

[10]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[11]

T. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4.  Google Scholar

[12]

L. L. SunB. Guo and W. J. Gao, A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., 37 (2014), 22-25.  doi: 10.1016/j.aml.2014.05.009.  Google Scholar

[13]

G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643.  doi: 10.4153/CJM-1980-049-5.  Google Scholar

[14]

Y. B. Yang and R. Z. Xu, Nonlinear wave equation with both strong and weakly damped terms: supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), 1351-1358.  doi: 10.3934/cpaa.2019065.  Google Scholar

[15]

J. Zhou, Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett., 45 (2015), 64-68.  doi: 10.1016/j.aml.2015.01.010.  Google Scholar

[16]

G. Zu and B. Guo, Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy, Evol. Equ. Control The., 10 (2021), 259-270.  doi: 10.3934/eect.2020065.  Google Scholar

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