Global existence for semilinear damped wave equations in relation with the Strauss conjecture

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February 2020, 40(2): 709-724. doi: 10.3934/dcds.2020058

Global existence for semilinear damped wave equations in relation with the Strauss conjecture

Mengyun Liu and Chengbo Wang ^,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

^* Corresponding author: Chengbo Wang

Received July 2018 Revised March 2019 Published November 2019

Fund Project: The authors were supported by NSFC 11971428.

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.

Keywords: Global existence, weighted strichartz estimates, damped wave equations, local energy estimates, Strauss conjecture.

Mathematics Subject Classification: Primary: 35L05, 35L15, 35L71; Secondary: 35B33, 58J45.

Citation: Mengyun Liu, Chengbo Wang. Global existence for semilinear damped wave equations in relation with the Strauss conjecture. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 709-724. doi: 10.3934/dcds.2020058

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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