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August  2020, 40(8): 4985-4999. doi: 10.3934/dcds.2020208

Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times

Wei Dai , Daoyuan Fang and  Chengbo Wang ,

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

* Corresponding author: Chengbo Wang

Received  October 2019 Revised  March 2020 Published  May 2020

Fund Project: The authors were supported by NSFC 11671353 and 11971428

By assuming certain local energy estimates on $ (1+3) $-dimensional asymptotically flat space-time, we study the existence portion of the Strauss type wave system. Firstly we give a kind of space-time estimates which are related to the local energy norm that appeared in [13]. These estimates can be used to prove a series of weighted Strichartz and KSS type estimates, for wave equations on asymptotically flat space-time. Then we apply the space-time estimates to obtain the lower bound of the lifespan when the nonlinear exponents $ p $ and $ q\ge 2 $. In particular, our bound for the subcritical case is sharp in general and we extend the known region of $ (p, q) $ to admit global solutions. In addition, the initial data are not required to be compactly supported, when $ p, q>2 $.

Keywords: Asymptotically flat space-time, Strauss conjecture, lifespan, weighted Strichartz estimates, local energy estimates.
Mathematics Subject Classification: 58J45, 35L71, 35L05, 35L15, 35B33.
Citation: Wei Dai, Daoyuan Fang, Chengbo Wang. Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4985-4999. doi: 10.3934/dcds.2020208
References:
[1]

R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for $p$-$q$ systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766.

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[3]

D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, In Geometrical optics and related topics (Cortona, 1996), volume 32 of Progr. Nonlinear Differential Equations Appl., pages 117a€"140. Birkhäuser Boston, Boston, MA, 1997.

[4]

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.

[5]

K. Hidano, J. Metcalfe, H. F. 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.

[6]

K. Hidano, C. Wang and K. Yokoyama, Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations, Math. Ann., 366 (2016), 667-694.  doi: 10.1007/s00208-015-1346-1.

[7]

J.-C. 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.

[8]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974.

[9]

T.-T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions, Indiana Univ. Math. J., 44 (1995), 1207-1248.  doi: 10.1512/iumj.1995.44.2026.

[10]

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.

[11]

J. Metcalfe and C. D. Sogge, Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal., 38 (2006), 188-209.  doi: 10.1137/050627149.

[12]

J. Metcalfe and D. Spencer, Global existence for a coupled wave system related to the Strauss conjecture, Commun. Pure Appl. Anal., 17 (2018), 593-604.  doi: 10.3934/cpaa.2018032.

[13]

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.

[14]

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.

[15]

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.

[16]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

[17]

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.

show all references

References:
[1]

R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for $p$-$q$ systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766.

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.

[3]

D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, In Geometrical optics and related topics (Cortona, 1996), volume 32 of Progr. Nonlinear Differential Equations Appl., pages 117a€"140. Birkhäuser Boston, Boston, MA, 1997.

[4]

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.

[5]

K. Hidano, J. Metcalfe, H. F. 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.

[6]

K. Hidano, C. Wang and K. Yokoyama, Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations, Math. Ann., 366 (2016), 667-694.  doi: 10.1007/s00208-015-1346-1.

[7]

J.-C. 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.

[8]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974.

[9]

T.-T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions, Indiana Univ. Math. J., 44 (1995), 1207-1248.  doi: 10.1512/iumj.1995.44.2026.

[10]

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.

[11]

J. Metcalfe and C. D. Sogge, Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal., 38 (2006), 188-209.  doi: 10.1137/050627149.

[12]

J. Metcalfe and D. Spencer, Global existence for a coupled wave system related to the Strauss conjecture, Commun. Pure Appl. Anal., 17 (2018), 593-604.  doi: 10.3934/cpaa.2018032.

[13]

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.

[14]

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.

[15]

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.

[16]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

[17]

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.

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