December  2019, 39(12): 7081-7099. doi: 10.3934/dcds.2019296

The Strauss conjecture on negatively curved backgrounds

1. 

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

2. 

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

* Corresponding author

To Luis Caffarelli on the occasion of his 70th birthday, with admiration and friendship

Received  November 2018 Revised  August 2019 Published  September 2019

Fund Project: The second author was supported in part by NSF Grant DMS-1665373 and the first two authors were supported in part by a Simons Fellowship. The third author was supported in part by NSFC 11971428 and National Support Program for Young Top-Notch Talents.

This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power $ p\in (1, 1+\frac{4}{n-1}] $ for the shifted wave equation on hyperbolic space $ {\mathbb{H}}^n $ involving nonlinearities of the form $ \pm |u|^p $ or $ \pm|u|^{p-1}u $. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge [9] (or Tataru [29]) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for the constant curvature case of Anker and Pierfelice [1] and Metcalfe and Taylor [22]. We also discuss the role of curvature and state a couple of open problems. Finally, in an appendix, we give an alternate proof of dispersive estimates of Tataru [29] for $ {\mathbb H}^3 $ and settle a dispute, in his favor, raised in [21] about his proof. Our proof is slightly more self-contained than the one in [29] since it does not make use of heavy spherical analysis on hyperbolic space such as the Harish-Chandra $ c $-function; instead it relies only on simple facts about Bessel potentials.

Citation: Yannick Sire, Christopher D. Sogge, Chengbo Wang. The Strauss conjecture on negatively curved backgrounds. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7081-7099. doi: 10.3934/dcds.2019296
References:
[1]

J.-P. Anker and V. Pierfelice, Wave and Klein-Gordon equations on hyperbolic spaces, Anal. PDE, 7 (2014), 953-995.   Google Scholar

[2]

J.-P. AnkerV. Pierfelice and M. Vallarino, The wave equation on hyperbolic spaces, J. Differential Equations, 252 (2012), 5613-5661.  doi: 10.1016/j.jde.2012.01.031.  Google Scholar

[3]

J.-P. AnkerV. Pierfelice and M. Vallarino, The wave equation on Damek-Ricci spaces, Ann. Mat. Pura Appl. (4), 194 (2015), 731-758.  doi: 10.1007/s10231-013-0395-x.  Google Scholar

[4]

N. Aronszajn and K. T. Smith, Theory of Bessel potentials. Ⅰ., Ann. Inst. Fourier, 11 (1961), 385-475.  doi: 10.5802/aif.116.  Google Scholar

[5]

A. Borbély, On the spectrum of the Laplacian in negatively curved manifolds, Studia Sci. Math. Hungar., 30 (1995), 375-378.   Google Scholar

[6]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.  Google Scholar

[7]

I. Chavel, Riemannian Geometry, volume 98 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 2006. A modern introduction. doi: 10.1017/CBO9780511616822.  Google Scholar

[8]

J. Fontaine, A semilinear wave equation on hyperbolic spaces, Comm. Partial Differential Equations, 22 (1997), 633-659.  doi: 10.1080/03605309708821277.  Google Scholar

[9]

V. GeorgievH. 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

[10]

R. T. Glassey, Existence in the large for $ \Box u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.  Google Scholar

[11]

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

[12]

C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.  doi: 10.1007/s00222-002-0268-1.  Google Scholar

[13]

E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.  Google Scholar

[14]

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

[15]

T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403.  Google Scholar

[16]

M. Keel and T. Tao, Small data blow-up for semilinear Klein-Gordon equations, Amer. J. Math., 121 (1999), 629-669.  doi: 10.1353/ajm.1999.0021.  Google Scholar

[17]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.  doi: 10.1353/ajm.1996.0042.  Google Scholar

[18]

H. Lindblad and C. D. Sogge, Restriction theorems and semilinear Klein-Gordon equations in (1+3)-dimensions, Duke Math. J., 85 (1996), 227-252.  doi: 10.1215/S0012-7094-96-08510-5.  Google Scholar

[19]

R. R. Mazzeo and R. B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., 75 (1987), 260-310.  doi: 10.1016/0022-1236(87)90097-8.  Google Scholar

[20]

H. P. McKean, An upper bound to the spectrum of $\Delta $ on a manifold of negative curvature, J. Differential Geometry, 4 (1970), 359-366.  doi: 10.4310/jdg/1214429509.  Google Scholar

[21]

J. Metcalfe and M. Taylor, Nonlinear waves on 3D hyperbolic space, Trans. Amer. Math. Soc., 363 (2011), 3489-3529.  doi: 10.1090/S0002-9947-2011-05122-6.  Google Scholar

[22]

J. Metcalfe and M. Taylor, Dispersive wave estimates on 3D hyperbolic space, Proc. Amer. Math. Soc., 140 (2012), 3861-3866.  doi: 10.1090/S0002-9939-2012-11534-5.  Google Scholar

[23]

J. Schaeffer, The equation $u_{tt}-\Delta u = |u|^p$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31-44.  doi: 10.1017/S0308210500026135.  Google Scholar

[24]

A. G. Setti, A lower bound for the spectrum of the Laplacian in terms of sectional and Ricci curvature, Proc. Amer. Math. Soc., 112 (1991), 277-282.  doi: 10.1090/S0002-9939-1991-1043421-3.  Google Scholar

[25]

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

[26]

C. D. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian, volume 188 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2014. doi: 10.1515/9781400850549.  Google Scholar

[27]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X.  Google Scholar

[28]

R.t S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[29]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795–807 (electronic). doi: 10.1090/S0002-9947-00-02750-1.  Google Scholar

[30]

M. Taylor, Partial Differential Equations Ⅱ. Qualitative Studies of Linear Equations. Second Edition, Applied Mathematical Sciences, 116. Springer, New York, 2011.  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, Recent works on the Strauss conjecture, In Recent Advances in Harmonic Analysis and Partial Differential Equations, volume 581 of Contemp. Math., pages 235–256, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/581/11497.  Google Scholar

[33]

B. T. Yordanov and Q. S. 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

[34]

Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations, 8 (1995), 135-144.   Google Scholar

[35]

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.-P. Anker and V. Pierfelice, Wave and Klein-Gordon equations on hyperbolic spaces, Anal. PDE, 7 (2014), 953-995.   Google Scholar

[2]

J.-P. AnkerV. Pierfelice and M. Vallarino, The wave equation on hyperbolic spaces, J. Differential Equations, 252 (2012), 5613-5661.  doi: 10.1016/j.jde.2012.01.031.  Google Scholar

[3]

J.-P. AnkerV. Pierfelice and M. Vallarino, The wave equation on Damek-Ricci spaces, Ann. Mat. Pura Appl. (4), 194 (2015), 731-758.  doi: 10.1007/s10231-013-0395-x.  Google Scholar

[4]

N. Aronszajn and K. T. Smith, Theory of Bessel potentials. Ⅰ., Ann. Inst. Fourier, 11 (1961), 385-475.  doi: 10.5802/aif.116.  Google Scholar

[5]

A. Borbély, On the spectrum of the Laplacian in negatively curved manifolds, Studia Sci. Math. Hungar., 30 (1995), 375-378.   Google Scholar

[6]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.  Google Scholar

[7]

I. Chavel, Riemannian Geometry, volume 98 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 2006. A modern introduction. doi: 10.1017/CBO9780511616822.  Google Scholar

[8]

J. Fontaine, A semilinear wave equation on hyperbolic spaces, Comm. Partial Differential Equations, 22 (1997), 633-659.  doi: 10.1080/03605309708821277.  Google Scholar

[9]

V. GeorgievH. 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

[10]

R. T. Glassey, Existence in the large for $ \Box u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.  Google Scholar

[11]

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

[12]

C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.  doi: 10.1007/s00222-002-0268-1.  Google Scholar

[13]

E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.  Google Scholar

[14]

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

[15]

T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403.  Google Scholar

[16]

M. Keel and T. Tao, Small data blow-up for semilinear Klein-Gordon equations, Amer. J. Math., 121 (1999), 629-669.  doi: 10.1353/ajm.1999.0021.  Google Scholar

[17]

H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.  doi: 10.1353/ajm.1996.0042.  Google Scholar

[18]

H. Lindblad and C. D. Sogge, Restriction theorems and semilinear Klein-Gordon equations in (1+3)-dimensions, Duke Math. J., 85 (1996), 227-252.  doi: 10.1215/S0012-7094-96-08510-5.  Google Scholar

[19]

R. R. Mazzeo and R. B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., 75 (1987), 260-310.  doi: 10.1016/0022-1236(87)90097-8.  Google Scholar

[20]

H. P. McKean, An upper bound to the spectrum of $\Delta $ on a manifold of negative curvature, J. Differential Geometry, 4 (1970), 359-366.  doi: 10.4310/jdg/1214429509.  Google Scholar

[21]

J. Metcalfe and M. Taylor, Nonlinear waves on 3D hyperbolic space, Trans. Amer. Math. Soc., 363 (2011), 3489-3529.  doi: 10.1090/S0002-9947-2011-05122-6.  Google Scholar

[22]

J. Metcalfe and M. Taylor, Dispersive wave estimates on 3D hyperbolic space, Proc. Amer. Math. Soc., 140 (2012), 3861-3866.  doi: 10.1090/S0002-9939-2012-11534-5.  Google Scholar

[23]

J. Schaeffer, The equation $u_{tt}-\Delta u = |u|^p$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31-44.  doi: 10.1017/S0308210500026135.  Google Scholar

[24]

A. G. Setti, A lower bound for the spectrum of the Laplacian in terms of sectional and Ricci curvature, Proc. Amer. Math. Soc., 112 (1991), 277-282.  doi: 10.1090/S0002-9939-1991-1043421-3.  Google Scholar

[25]

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

[26]

C. D. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian, volume 188 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2014. doi: 10.1515/9781400850549.  Google Scholar

[27]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X.  Google Scholar

[28]

R.t S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[29]

D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795–807 (electronic). doi: 10.1090/S0002-9947-00-02750-1.  Google Scholar

[30]

M. Taylor, Partial Differential Equations Ⅱ. Qualitative Studies of Linear Equations. Second Edition, Applied Mathematical Sciences, 116. Springer, New York, 2011.  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, Recent works on the Strauss conjecture, In Recent Advances in Harmonic Analysis and Partial Differential Equations, volume 581 of Contemp. Math., pages 235–256, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/conm/581/11497.  Google Scholar

[33]

B. T. Yordanov and Q. S. 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

[34]

Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations, 8 (1995), 135-144.   Google Scholar

[35]

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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