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Positivity, monotonicity, and convexity for convolution operators
Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
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 [
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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