American Institute of Mathematical Sciences

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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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) Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[22] C. D. Sogge, Lectures on Non-linear Wave Equations, International Press, Boston, MA, second edition, 2008.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. Wang, Recent progress on the strauss conjecture and related problems., SCIENTIA SINICA Mathematica, 48 (2018), 111-130.   Google Scholar

[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.  Google Scholar

[33]

C. Wang and X. Yu, Recent works on the Strauss conjecture, Contemp. Math., 581 (2012), 235-256.  doi: 10.1090/conm/581/11497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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) Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22] C. D. Sogge, Lectures on Non-linear Wave Equations, International Press, Boston, MA, second edition, 2008.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. Wang, Recent progress on the strauss conjecture and related problems., SCIENTIA SINICA Mathematica, 48 (2018), 111-130.   Google Scholar

[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.  Google Scholar

[33]

C. Wang and X. Yu, Recent works on the Strauss conjecture, Contemp. Math., 581 (2012), 235-256.  doi: 10.1090/conm/581/11497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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