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On mean field systems with multi-classes
Global existence for semilinear damped wave equations in relation with the Strauss conjecture
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $ n $ dimensional nontrapping asymptotically Euclidean manifolds, when $ n = 3, 4 $ as well as two dimensional Euclidean space. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.
References:
[1] |
J. Bony and D. Häfner,
The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations, 35 (2010), 23-67.
doi: 10.1080/03605300903396601. |
[2] |
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. |
[3] |
D. Fang and C. Wang,
Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.
doi: 10.1515/FORM.2011.009. |
[4] |
K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, Funkcial. Ekvac., 62 (2019), 157-189. |
[5] |
V. Georgiev, H. Lindblad and C. D. Sogge,
Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.
doi: 10.1353/ajm.1997.0038. |
[6] |
R. T. Glassey,
Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.
doi: 10.1007/BF01162066. |
[7] |
K. Hidano, J. Metcalfe, H. Smith, C. D. Sogge and Y. Zhou,
On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809.
doi: 10.1090/S0002-9947-09-05053-3. |
[8] |
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. |
[9] |
J. Jiang, C. Wang and X. Yu,
Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal., 11 (2012), 1723-1752.
doi: 10.3934/cpaa.2012.11.1723. |
[10] |
F. John and S. Klainerman,
Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.
doi: 10.1002/cpa.3160370403. |
[11] |
F. John,
Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.
doi: 10.1007/BF01647974. |
[12] |
N. Lai and H. Takamura,
Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case, Nonlinear Anal., 168 (2018), 222-237.
doi: 10.1016/j.na.2017.12.008. |
[13] |
Ni. 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. Lai and Y. Zhou,
The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions, J. Math. Pures Appl., 123 (2019), 229-243.
doi: 10.1016/j.matpur.2018.04.009. |
[15] |
T. Li and Y. Zhou,
Breakdown of solutions to $\square u+u_t = |u|^{1+\alpha}$, Discrete Contin. Dynam. Systems, 1 (1995), 503-520.
doi: 10.3934/dcds.1995.1.503. |
[16] |
J. Lin, K. Nishihara and J. Zhai,
Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst., 32 (2012), 4307-4320.
doi: 10.3934/dcds.2012.32.4307. |
[17] |
H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and C. Wang,
The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661.
doi: 10.1007/s00208-014-1006-x. |
[18] |
J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math., preprint, arXiv: 1703.08064, (2017) |
[19] |
J. Metcalfe and D. Tataru,
Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353 (2012), 1183-1237.
doi: 10.1007/s00208-011-0714-8. |
[20] |
J. Metcalfe and C. Wang,
The Strauss conjecture on asymptotically flat space-times, SIAM J. Math. Anal., 49 (2017), 4579-4594.
doi: 10.1137/16M1074886. |
[21] |
T. C. 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. |
[22] |
C. D. Sogge, Lectures on Non-linear Wave Equations, International Press, Boston, MA, second edition, 2008.
![]() ![]() |
[23] |
C. D. Sogge and C. Wang,
Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.
doi: 10.1007/s11854-010-0023-2. |
[24] |
D. Tataru,
Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.
doi: 10.1090/S0002-9947-00-02750-1. |
[25] |
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. |
[26] |
Z. Tu and J. Lin,
Life-span of semilinear wave equations with scale-invariant damping: Critical strauss exponent case, Differential Integral Equations, 32 (2019), 249-264.
|
[27] |
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. |
[28] |
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. |
[29] |
C. Wang,
The Glassey conjecture on asymptotically flat manifolds, Trans. Amer. Math. Soc., 367 (2015), 7429-7451.
doi: 10.1090/S0002-9947-2014-06423-4. |
[30] |
C. Wang,
Long-time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174.
doi: 10.1080/03605302.2017.1345939. |
[31] |
C. Wang,
Recent progress on the strauss conjecture and related problems., SCIENTIA SINICA Mathematica, 48 (2018), 111-130.
|
[32] |
C. Wang and X. Yu,
Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.
doi: 10.1016/j.jmaa.2011.01.053. |
[33] |
C. Wang and X. Yu,
Recent works on the Strauss conjecture, Contemp. Math., 581 (2012), 235-256.
doi: 10.1090/conm/581/11497. |
[34] |
C. Wang and X. Yu,
Global existence of null-form wave equations on small asymptotically Euclidean manifolds, J. Funct. Anal., 266 (2014), 5676-5708.
doi: 10.1016/j.jfa.2014.02.028. |
[35] |
S. Yang,
Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal., 209 (2013), 683-728.
doi: 10.1007/s00205-013-0631-y. |
[36] |
S. Yang,
On the quasilinear wave equations in time dependent inhomogeneous media, J. Hyperbolic Differ. Equ., 13 (2016), 273-330.
doi: 10.1142/S0219891616500090. |
[37] |
B. Yordanov and Q. 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. |
[38] |
Y. Zhou,
Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.
doi: 10.1007/s11401-005-0205-x. |
show all references
References:
[1] |
J. Bony and D. Häfner,
The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations, 35 (2010), 23-67.
doi: 10.1080/03605300903396601. |
[2] |
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. |
[3] |
D. Fang and C. Wang,
Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.
doi: 10.1515/FORM.2011.009. |
[4] |
K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, Funkcial. Ekvac., 62 (2019), 157-189. |
[5] |
V. Georgiev, H. Lindblad and C. D. Sogge,
Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.
doi: 10.1353/ajm.1997.0038. |
[6] |
R. T. Glassey,
Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.
doi: 10.1007/BF01162066. |
[7] |
K. Hidano, J. Metcalfe, H. Smith, C. D. Sogge and Y. Zhou,
On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809.
doi: 10.1090/S0002-9947-09-05053-3. |
[8] |
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. |
[9] |
J. Jiang, C. Wang and X. Yu,
Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal., 11 (2012), 1723-1752.
doi: 10.3934/cpaa.2012.11.1723. |
[10] |
F. John and S. Klainerman,
Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.
doi: 10.1002/cpa.3160370403. |
[11] |
F. John,
Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.
doi: 10.1007/BF01647974. |
[12] |
N. Lai and H. Takamura,
Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case, Nonlinear Anal., 168 (2018), 222-237.
doi: 10.1016/j.na.2017.12.008. |
[13] |
Ni. 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. Lai and Y. Zhou,
The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions, J. Math. Pures Appl., 123 (2019), 229-243.
doi: 10.1016/j.matpur.2018.04.009. |
[15] |
T. Li and Y. Zhou,
Breakdown of solutions to $\square u+u_t = |u|^{1+\alpha}$, Discrete Contin. Dynam. Systems, 1 (1995), 503-520.
doi: 10.3934/dcds.1995.1.503. |
[16] |
J. Lin, K. Nishihara and J. Zhai,
Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst., 32 (2012), 4307-4320.
doi: 10.3934/dcds.2012.32.4307. |
[17] |
H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and C. Wang,
The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661.
doi: 10.1007/s00208-014-1006-x. |
[18] |
J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math., preprint, arXiv: 1703.08064, (2017) |
[19] |
J. Metcalfe and D. Tataru,
Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353 (2012), 1183-1237.
doi: 10.1007/s00208-011-0714-8. |
[20] |
J. Metcalfe and C. Wang,
The Strauss conjecture on asymptotically flat space-times, SIAM J. Math. Anal., 49 (2017), 4579-4594.
doi: 10.1137/16M1074886. |
[21] |
T. C. 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. |
[22] |
C. D. Sogge, Lectures on Non-linear Wave Equations, International Press, Boston, MA, second edition, 2008.
![]() ![]() |
[23] |
C. D. Sogge and C. Wang,
Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.
doi: 10.1007/s11854-010-0023-2. |
[24] |
D. Tataru,
Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807.
doi: 10.1090/S0002-9947-00-02750-1. |
[25] |
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. |
[26] |
Z. Tu and J. Lin,
Life-span of semilinear wave equations with scale-invariant damping: Critical strauss exponent case, Differential Integral Equations, 32 (2019), 249-264.
|
[27] |
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. |
[28] |
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. |
[29] |
C. Wang,
The Glassey conjecture on asymptotically flat manifolds, Trans. Amer. Math. Soc., 367 (2015), 7429-7451.
doi: 10.1090/S0002-9947-2014-06423-4. |
[30] |
C. Wang,
Long-time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174.
doi: 10.1080/03605302.2017.1345939. |
[31] |
C. Wang,
Recent progress on the strauss conjecture and related problems., SCIENTIA SINICA Mathematica, 48 (2018), 111-130.
|
[32] |
C. Wang and X. Yu,
Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.
doi: 10.1016/j.jmaa.2011.01.053. |
[33] |
C. Wang and X. Yu,
Recent works on the Strauss conjecture, Contemp. Math., 581 (2012), 235-256.
doi: 10.1090/conm/581/11497. |
[34] |
C. Wang and X. Yu,
Global existence of null-form wave equations on small asymptotically Euclidean manifolds, J. Funct. Anal., 266 (2014), 5676-5708.
doi: 10.1016/j.jfa.2014.02.028. |
[35] |
S. Yang,
Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal., 209 (2013), 683-728.
doi: 10.1007/s00205-013-0631-y. |
[36] |
S. Yang,
On the quasilinear wave equations in time dependent inhomogeneous media, J. Hyperbolic Differ. Equ., 13 (2016), 273-330.
doi: 10.1142/S0219891616500090. |
[37] |
B. Yordanov and Q. 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. |
[38] |
Y. Zhou,
Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.
doi: 10.1007/s11401-005-0205-x. |
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