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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 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 & Control Theory, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[5]

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

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[19]

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

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

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

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

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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