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

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 $.

Citation: Wei Dai, Daoyuan Fang, Chengbo Wang. Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4985-4999. doi: 10.3934/dcds.2020208
References:
[1]

R. AgemiY. 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

K. HidanoJ. MetcalfeH. F. SmithC. 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

[6]

K. HidanoC. 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.  Google Scholar

[7]

J.-C. JiangC. 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

[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.  Google Scholar

[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.  Google Scholar

[10]

H. LindbladJ. MetcalfeC. D. SoggeM. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

show all references

References:
[1]

R. AgemiY. 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

K. HidanoJ. MetcalfeH. F. SmithC. 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

[6]

K. HidanoC. 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.  Google Scholar

[7]

J.-C. JiangC. 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

[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.  Google Scholar

[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.  Google Scholar

[10]

H. LindbladJ. MetcalfeC. D. SoggeM. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[1]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[2]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[3]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[4]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[5]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[6]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[7]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[8]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[9]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[10]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[12]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[13]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[14]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[15]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[16]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[17]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[18]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[19]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[20]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (66)
  • HTML views (99)
  • Cited by (0)

Other articles
by authors

[Back to Top]