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Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling
Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass
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 |
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 [
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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