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April  2022, 11(2): 515-536. doi: 10.3934/eect.2021011

Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass

Masahiro Ikeda 1, , Ziheng Tu 2,, and  Kyouhei Wakasa 3,

1. 

Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan, Center for Advanced Intelligence Project, RIKEN, Japan
2. 

Department of Mathematics, School of Data Science, Zhejiang University of Finance and Economics, 310018, Hangzhou, P.R.China
3. 

Department of Creative Engineering, National Institute of Technology, Kushiro College, 2-32-1 Otanoshike-Nishi, Kushiro-Shi, Hokkaido 084-0916, Japan

* Corresponding author: Ziheng Tu

Received  March 2020 Revised  December 2020 Published  April 2022 Early access  March 2021

In the present paper, we study small data blow-up of the semi-linear wave equation with a scattering dissipation term and a time-dependent mass term from the aspect of wave-like behavior. The Strauss type critical exponent is determined and blow-up results are obtained to both sub-critical and critical cases with corresponding upper bound lifespan estimates. For the sub-critical case, our argument does not rely on the sign condition of dissipation and mass, which gives the extension of the result in [14]. Moreover, we show the blow-up result for the critical case which is a new result.

Keywords: Semi-linear, damped wave equation, Strauss exponent, blow-up, lifespan.
Mathematics Subject Classification: Primary: 35L71; Secondary: 35B44.
Citation: Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011
References:
[1]

M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Math. Methods in Appl. Sci., 38 (2015), 1032-1045.  doi: 10.1002/mma.3126.

[2]

M. D'Abbicco, S. Lucente and M. Reissig, Semi-linear wave equations with effective damping, Chin. Ann. Math., 34 (2013), 345-380.  doi: 10.1007/s11401-013-0773-0.

[3]

M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 (2015), 5040-5073.  doi: 10.1016/j.jde.2015.06.018.

[4]

M. D'Abbicco, G. Girardi and M. Reissig, A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Anal., 179 (2015), 15-40.  doi: 10.1016/j.na.2018.08.006.

[5]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Math Society, 2010. doi: 10.1090/gsm/019.

[6]

K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimate of lifespan and blow-up rates for the semilinear wave equation with time-dependent damping and subcritical nonlinearities, Funkcial. Ekvac., 62 (2019), 5165-5201.  doi: 10.1619/fesi.62.157.

[7]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.

[8]

M. Ikeda and T. Inui, The sharp estimate of the lifespan for the semilinear wave equation with time-dependent damping, Diff. Int. Equs., 32 (2019), 1-36. 

[9]

M. Ikeda and M. Sobajima, Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data, Math. Ann., 372 (2018), 1017-1040.  doi: 10.1007/s00208-018-1664-1.

[10]

M. Ikeda, M. Sobajima and Y. Wakasugi, Sharp lifespan estimates of blowup solutions to semilinear wave equations with time-dependent effective damping, J. Hyperbolic Differential Equations, 16 (2019), 495-517.  doi: 10.1142/S0219891619500176.

[11]

M. Ikeda and Y. Wakasugi, Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case, Proc. Amer. Math. Soc., 148 (2020), 157-172.  doi: 10.1090/proc/14297.

[12]

N.-A. Lai and H. Takamura, Blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case, Nonlinear Anal. TMA, 168 (2018), 222-237.  doi: 10.1016/j.na.2017.12.008.

[13]

N.-A. Lai, H. Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377-5394.  doi: 10.1016/j.jde.2017.06.017.

[14]

N.-A. Lai, N. M. Schiavone and H. Takamura, Wave-like blow-up for semilinear wave equations with scattering damping and negative mass, in New Tools for Nonlinear PDEs and Application, Trends in Mathematics, Birkhäuser, (2019), 217–240. doi: 10.1007/978-3-030-10937-0_8.

[15]

N.-A. Lai, N. M. Schiavone and H. Takamura, Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping, to appear in Advanced Studies in Pure Mathematics.

[16]

J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discr. Cont. Dyn. Syst.- Series A, 32 (2012), 4307-4320.  doi: 10.3934/dcds.2012.32.4307.

[17]

M. Liu and C. Wang, Global existence of semilinear damped wave equations in relation with the Strauss conjecture, Discr. Cont. Dyn. Syst.- Series A, 40 (2020), 709-724.  doi: 10.3934/dcds.2020058.

[18]

A. Palmieri and Z. Tu, Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity, J. Math. Anal. Appl., 470 (2019), 447-469.  doi: 10.1016/j.jmaa.2018.10.015.

[19]

M. Struwe, Semilinear wave equations, Bulletin of the American Mathematical Society, 26 (1992), 53-85.  doi: 10.1090/S0273-0979-1992-00225-2.

[20] C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Mathematics in Science and Engineering, 48, Academic Press, New York-London, 1968. 
[21]

H. Takamura, Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations, Nonlinear Anal. TMA, 125 (2015), 227-240.  doi: 10.1016/j.na.2015.05.024.

[22]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.

[23]

Z. Tu and J. Lin, A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent, preprint, arXiv: 1709.00866.

[24]

Z. Tu and J. Lin, Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case, Diff. Int. Equs., 32 (2019), 249-264. 

[25]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differential Equations, 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[26]

K. Wakasa and B. Yordanov, On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case, Nonlinear Anal., 180 (2019), 67-74.  doi: 10.1016/j.na.2018.09.012.

[27]

Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, in Fourier analysis, Trends Math., Birkhäuser/Springer, Cham, (2014), 375–390.

[28]

Y. Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452-487.  doi: 10.1016/j.jmaa.2016.10.018.

[29]

J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Methods Appl. Sci., 27 (2004), 101-124.  doi: 10.1002/mma.446.

[30]

J. Wirth, Wave equations with time-dependent dissipation. Ⅰ. Non-effective dissipation, J. Differential Equations, 222 (2006), 487-514.  doi: 10.1016/j.jde.2005.07.019.

[31]

J. Wirth, Wave equations with time-dependent dissipation. Ⅱ. Effective dissipation, J. Differential Equations, 232 (2007), 74-103.  doi: 10.1016/j.jde.2006.06.004.

[32]

B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012.

[33]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Math. Acad. Sci. Paris, Sér. I, 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

show all references

References:
[1]

M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Math. Methods in Appl. Sci., 38 (2015), 1032-1045.  doi: 10.1002/mma.3126.

[2]

M. D'Abbicco, S. Lucente and M. Reissig, Semi-linear wave equations with effective damping, Chin. Ann. Math., 34 (2013), 345-380.  doi: 10.1007/s11401-013-0773-0.

[3]

M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 (2015), 5040-5073.  doi: 10.1016/j.jde.2015.06.018.

[4]

M. D'Abbicco, G. Girardi and M. Reissig, A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Anal., 179 (2015), 15-40.  doi: 10.1016/j.na.2018.08.006.

[5]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Math Society, 2010. doi: 10.1090/gsm/019.

[6]

K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimate of lifespan and blow-up rates for the semilinear wave equation with time-dependent damping and subcritical nonlinearities, Funkcial. Ekvac., 62 (2019), 5165-5201.  doi: 10.1619/fesi.62.157.

[7]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.

[8]

M. Ikeda and T. Inui, The sharp estimate of the lifespan for the semilinear wave equation with time-dependent damping, Diff. Int. Equs., 32 (2019), 1-36. 

[9]

M. Ikeda and M. Sobajima, Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data, Math. Ann., 372 (2018), 1017-1040.  doi: 10.1007/s00208-018-1664-1.

[10]

M. Ikeda, M. Sobajima and Y. Wakasugi, Sharp lifespan estimates of blowup solutions to semilinear wave equations with time-dependent effective damping, J. Hyperbolic Differential Equations, 16 (2019), 495-517.  doi: 10.1142/S0219891619500176.

[11]

M. Ikeda and Y. Wakasugi, Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case, Proc. Amer. Math. Soc., 148 (2020), 157-172.  doi: 10.1090/proc/14297.

[12]

N.-A. Lai and H. Takamura, Blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case, Nonlinear Anal. TMA, 168 (2018), 222-237.  doi: 10.1016/j.na.2017.12.008.

[13]

N.-A. Lai, H. Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377-5394.  doi: 10.1016/j.jde.2017.06.017.

[14]

N.-A. Lai, N. M. Schiavone and H. Takamura, Wave-like blow-up for semilinear wave equations with scattering damping and negative mass, in New Tools for Nonlinear PDEs and Application, Trends in Mathematics, Birkhäuser, (2019), 217–240. doi: 10.1007/978-3-030-10937-0_8.

[15]

N.-A. Lai, N. M. Schiavone and H. Takamura, Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping, to appear in Advanced Studies in Pure Mathematics.

[16]

J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discr. Cont. Dyn. Syst.- Series A, 32 (2012), 4307-4320.  doi: 10.3934/dcds.2012.32.4307.

[17]

M. Liu and C. Wang, Global existence of semilinear damped wave equations in relation with the Strauss conjecture, Discr. Cont. Dyn. Syst.- Series A, 40 (2020), 709-724.  doi: 10.3934/dcds.2020058.

[18]

A. Palmieri and Z. Tu, Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity, J. Math. Anal. Appl., 470 (2019), 447-469.  doi: 10.1016/j.jmaa.2018.10.015.

[19]

M. Struwe, Semilinear wave equations, Bulletin of the American Mathematical Society, 26 (1992), 53-85.  doi: 10.1090/S0273-0979-1992-00225-2.

[20] C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Mathematics in Science and Engineering, 48, Academic Press, New York-London, 1968. 
[21]

H. Takamura, Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations, Nonlinear Anal. TMA, 125 (2015), 227-240.  doi: 10.1016/j.na.2015.05.024.

[22]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489.  doi: 10.1006/jdeq.2000.3933.

[23]

Z. Tu and J. Lin, A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent, preprint, arXiv: 1709.00866.

[24]

Z. Tu and J. Lin, Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case, Diff. Int. Equs., 32 (2019), 249-264. 

[25]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differential Equations, 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[26]

K. Wakasa and B. Yordanov, On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case, Nonlinear Anal., 180 (2019), 67-74.  doi: 10.1016/j.na.2018.09.012.

[27]

Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, in Fourier analysis, Trends Math., Birkhäuser/Springer, Cham, (2014), 375–390.

[28]

Y. Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients, J. Math. Anal. Appl., 447 (2017), 452-487.  doi: 10.1016/j.jmaa.2016.10.018.

[29]

J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Methods Appl. Sci., 27 (2004), 101-124.  doi: 10.1002/mma.446.

[30]

J. Wirth, Wave equations with time-dependent dissipation. Ⅰ. Non-effective dissipation, J. Differential Equations, 222 (2006), 487-514.  doi: 10.1016/j.jde.2005.07.019.

[31]

J. Wirth, Wave equations with time-dependent dissipation. Ⅱ. Effective dissipation, J. Differential Equations, 232 (2007), 74-103.  doi: 10.1016/j.jde.2006.06.004.

[32]

B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012.

[33]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Math. Acad. Sci. Paris, Sér. I, 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

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