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 and Continuous Dynamical Systems, 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. 

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

M. Taylor, Partial Differential Equations Ⅱ. Qualitative Studies of Linear Equations. Second Edition, Applied Mathematical Sciences, 116. Springer, New York, 2011.

[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, 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.

[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.

[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. 

[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.

show all references

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

References:
[1]

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

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

M. Taylor, Partial Differential Equations Ⅱ. Qualitative Studies of Linear Equations. Second Edition, Applied Mathematical Sciences, 116. Springer, New York, 2011.

[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, 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.

[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.

[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. 

[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.

[1]

Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723

[2]

Seongyeon Kim, Yehyun Kwon, Ihyeok Seo. Strichartz estimates and local regularity for the elastic wave equation with singular potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1897-1911. doi: 10.3934/dcds.2020344

[3]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210

[4]

Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134

[5]

Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143

[6]

Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177

[7]

Gong Chen. Strichartz estimates for charge transfer models. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050

[8]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100

[9]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

[10]

Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771

[11]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233

[12]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109

[13]

Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure and Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031

[14]

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

[15]

Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024

[16]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46

[17]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801

[18]

Fabio Nicola. Remarks on dispersive estimates and curvature. Communications on Pure and Applied Analysis, 2007, 6 (1) : 203-212. doi: 10.3934/cpaa.2007.6.203

[19]

Jason Metcalfe, David Spencer. Global existence for a coupled wave system related to the Strauss conjecture. Communications on Pure and Applied Analysis, 2018, 17 (2) : 593-604. doi: 10.3934/cpaa.2018032

[20]

Sun-Yung Alice Chang, Xi-Nan Ma, Paul Yang. Principal curvature estimates for the convex level sets of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1151-1164. doi: 10.3934/dcds.2010.28.1151

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (220)
  • HTML views (158)
  • Cited by (0)

[Back to Top]