## The Glassey conjecture on asymptotically flat manifolds

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- by Chengbo Wang PDF
- Trans. Amer. Math. Soc.
**367**(2015), 7429-7451 Request permission

## Abstract:

We verify the $3$-dimensional Glassey conjecture on asymptotically flat manifolds $(\mathbb {R}^{1+3},\mathfrak {g})$, where the metric $\mathfrak {g}$ is a certain small space-time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds. Moreover, for radial asymptotically flat manifolds $(\mathbb {R}^{1+n},\mathfrak {g})$ with $n\ge 3$, we verify the Glassey conjecture in the radial case. High dimensional wave equations with higher regularity are also discussed. The main idea is to exploit local energy and KSS estimates with variable coefficients, together with the weighted Sobolev estimates including trace estimates.## References

- Serge Alinhac,
*On the Morawetz–Keel-Smith-Sogge inequality for the wave equation on a curved background*, Publ. Res. Inst. Math. Sci.**42**(2006), no. 3, 705–720. MR**2266993**, DOI 10.2977/prims/1166642156 - Jean-François Bony and Dietrich Häfner,
*The semilinear wave equation on asymptotically Euclidean manifolds*, Comm. Partial Differential Equations**35**(2010), no. 1, 23–67. MR**2748617**, DOI 10.1080/03605300903396601 - N. Burq,
*Global Strichartz estimates for nontrapping geometries: about an article by H. F. Smith and C. D. Sogge: “Global Strichartz estimates for nontrapping perturbations of the Laplacian” [Comm. Partial Differential Equation 25 (2000), no. 11-12 2171–2183; MR1789924 (2001j:35180)]*, Comm. Partial Differential Equations**28**(2003), no. 9-10, 1675–1683. MR**2001179**, DOI 10.1081/PDE-120024528 - Daoyuan Fang and Chengbo Wang,
*Weighted Strichartz estimates with angular regularity and their applications*, Forum Math.**23**(2011), no. 1, 181–205. MR**2769870**, DOI 10.1515/FORM.2011.009 - Daoyuan Fang and Chengbo Wang,
*Almost global existence for some semilinear wave equations with almost critical regularity*, Comm. Partial Differential Equations**38**(2013), no. 9, 1467–1491. MR**3169752**, DOI 10.1080/03605302.2013.803482 - R. T. Glassey, MathReview to “Global behavior of solutions to nonlinear wave equations in three space dimensions” of Sideris, Comm. Partial Differential Equations (1983).
- Kunio Hidano, Jason Metcalfe, Hart F. Smith, Christopher D. Sogge, and Yi Zhou,
*On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles*, Trans. Amer. Math. Soc.**362**(2010), no. 5, 2789–2809. MR**2584618**, DOI 10.1090/S0002-9947-09-05053-3 - Kunio Hidano and Kimitoshi Tsutaya,
*Global existence and asymptotic behavior of solutions for nonlinear wave equations*, Indiana Univ. Math. J.**44**(1995), no. 4, 1273–1305. MR**1386769**, DOI 10.1512/iumj.1995.44.2028 - Kunio Hidano, Chengbo Wang, and Kazuyoshi Yokoyama,
*On almost global existence and local well posedness for some 3-D quasi-linear wave equations*, Adv. Differential Equations**17**(2012), no. 3-4, 267–306. MR**2919103** - Kunio Hidano, Chengbo Wang, and Kazuyoshi Yokoyama,
*The Glassey conjecture with radially symmetric data*, J. Math. Pures Appl. (9)**98**(2012), no. 5, 518–541 (English, with English and French summaries). MR**2980460**, DOI 10.1016/j.matpur.2012.01.007 - Kunio Hidano and Kazuyoshi Yokoyama,
*A remark on the almost global existence theorems of Keel, Smith and Sogge*, Funkcial. Ekvac.**48**(2005), no. 1, 1–34. MR**2154375**, DOI 10.1619/fesi.48.1 - Markus Keel, Hart F. Smith, and Christopher D. Sogge,
*Almost global existence for some semilinear wave equations*, J. Anal. Math.**87**(2002), 265–279. Dedicated to the memory of Thomas H. Wolff. MR**1945285**, DOI 10.1007/BF02868477 - Sergiu Klainerman,
*Uniform decay estimates and the Lorentz invariance of the classical wave equation*, Comm. Pure Appl. Math.**38**(1985), no. 3, 321–332. MR**784477**, DOI 10.1002/cpa.3160380305 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*On the Zakharov and Zakharov-Schulman systems*, J. Funct. Anal.**127**(1995), no. 1, 204–234. MR**1308623**, DOI 10.1006/jfan.1995.1009 - H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu, C. Wang
*The Strauss conjecture on Kerr black hole backgrounds*. Math. Ann., to appear, DOI 10.1007/s00208-014-1006-x. arXiv 1304.4145 - Jason Metcalfe and Christopher D. Sogge,
*Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods*, SIAM J. Math. Anal.**38**(2006), no. 1, 188–209. MR**2217314**, DOI 10.1137/050627149 - Jason Metcalfe and Daniel Tataru,
*Global parametrices and dispersive estimates for variable coefficient wave equations*, Math. Ann.**353**(2012), no. 4, 1183–1237. MR**2944027**, DOI 10.1007/s00208-011-0714-8 - Jason Metcalfe, Daniel Tataru, and Mihai Tohaneanu,
*Price’s law on nonstationary space-times*, Adv. Math.**230**(2012), no. 3, 995–1028. MR**2921169**, DOI 10.1016/j.aim.2012.03.010 - Cathleen S. Morawetz,
*Time decay for the nonlinear Klein-Gordon equations*, Proc. Roy. Soc. London Ser. A**306**(1968), 291–296. MR**234136**, DOI 10.1098/rspa.1968.0151 - M. A. Rammaha,
*Finite-time blow-up for nonlinear wave equations in high dimensions*, Comm. Partial Differential Equations**12**(1987), no. 6, 677–700. MR**879355**, DOI 10.1080/03605308708820506 - Jack Schaeffer,
*Finite-time blow-up for $u_{tt}-\Delta u=H(u_r,u_t)$*, Comm. Partial Differential Equations**11**(1986), no. 5, 513–543. MR**829595**, DOI 10.1080/03605308608820434 - Thomas C. Sideris,
*Global behavior of solutions to nonlinear wave equations in three dimensions*, Comm. Partial Differential Equations**8**(1983), no. 12, 1291–1323. MR**711440**, DOI 10.1080/03605308308820304 - Hart F. Smith and Christopher D. Sogge,
*Global Strichartz estimates for nontrapping perturbations of the Laplacian*, Comm. Partial Differential Equations**25**(2000), no. 11-12, 2171–2183. MR**1789924**, DOI 10.1080/03605300008821581 - Christopher D. Sogge and Chengbo Wang,
*Concerning the wave equation on asymptotically Euclidean manifolds*, J. Anal. Math.**112**(2010), 1–32. MR**2762995**, DOI 10.1007/s11854-010-0023-2 - E. M. Stein and Guido Weiss,
*Fractional integrals on $n$-dimensional Euclidean space*, J. Math. Mech.**7**(1958), 503–514. MR**0098285**, DOI 10.1512/iumj.1958.7.57030 - Jacob Sterbenz,
*Angular regularity and Strichartz estimates for the wave equation*, Int. Math. Res. Not.**4**(2005), 187–231. With an appendix by Igor Rodnianski. MR**2128434**, DOI 10.1155/IMRN.2005.187 - Walter A. Strauss,
*Dispersal of waves vanishing on the boundary of an exterior domain*, Comm. Pure Appl. Math.**28**(1975), 265–278. MR**367461**, DOI 10.1002/cpa.3160280205 - Daniel Tataru,
*Local decay of waves on asymptotically flat stationary space-times*, Amer. J. Math.**135**(2013), no. 2, 361–401. MR**3038715**, DOI 10.1353/ajm.2013.0012 - Nickolay Tzvetkov,
*Existence of global solutions to nonlinear massless Dirac system and wave equation with small data*, Tsukuba J. Math.**22**(1998), no. 1, 193–211. MR**1637692**, DOI 10.21099/tkbjm/1496163480 - Chengbo Wang and Xin Yu,
*Concerning the Strauss conjecture on asymptotically Euclidean manifolds*, J. Math. Anal. Appl.**379**(2011), no. 2, 549–566. MR**2784342**, DOI 10.1016/j.jmaa.2011.01.053 - Chengbo Wang and Xin Yu,
*Global existence of null-form wave equations on small asymptotically Euclidean manifolds*, J. Funct. Anal.**266**(2014), no. 9, 5676–5708. MR**3182955**, DOI 10.1016/j.jfa.2014.02.028 - Yi Zhou,
*Blow up of solutions to the Cauchy problem for nonlinear wave equations*, Chinese Ann. Math. Ser. B**22**(2001), no. 3, 275–280. MR**1845748**, DOI 10.1142/S0252959901000280 - Yi Zhou and Wei Han,
*Blow-up of solutions to semilinear wave equations with variable coefficients and boundary*, J. Math. Anal. Appl.**374**(2011), no. 2, 585–601. MR**2729246**, DOI 10.1016/j.jmaa.2010.08.052

## Additional Information

**Chengbo Wang**- Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
- MR Author ID: 766167
- ORCID: 0000-0002-4878-7629
- Email: wangcbo@gmail.com
- Received by editor(s): June 28, 2013
- Received by editor(s) in revised form: March 10, 2014
- Published electronically: September 23, 2014
- Additional Notes: The author was supported by the Zhejiang Provincial Natural Science Foundation of China LR12A01002, the Fundamental Research Funds for the Central Universities, NSFC 11301478, 11271322 and J1210038.
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 7429-7451 - MSC (2010): Primary 35L70, 35L15
- DOI: https://doi.org/10.1090/S0002-9947-2014-06423-4
- MathSciNet review: 3378835