June  2022, 11(3): 635-648. 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 Published  June 2022 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 and Control Theory, 2022, 11 (3) : 635-648. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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