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March 2018, 17(2): 593-604. doi: 10.3934/cpaa.2018032

Global existence for a coupled wave system related to the Strauss conjecture

1. 

Department of Mathematics, University of North Carolina -Chapel Hill, Chapel Hill, NC, 27599-3250, USA

2. 

UCLA Mathematics Department, Box 951555, Los Angeles, CA, 90095-1555, USA

* Corresponding author: David Spencer

Received  March 2017 Revised  August 2017 Published  March 2018

Fund Project: The first author was supported in part by NSF grant DMS-1054289. The second author was supported in part by a Summer Undergraduate Research Fellowship (SURF) through the University of North Carolina, and the results contained herein were developed as a part of his Undergraduate Honors Thesis

A coupled system of semilinear wave equations is considered, and a small data global existence result related to the Strauss conjecture is proved. Previous results have shown that one of the powers may be reduced below the critical power for the Strauss conjecture provided the other power sufficiently exceeds such. The stability of such results under asymptotically flat perturbations of the space-time where an integrated local energy decay estimate is available is established.

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

[2]

S. Alinhac, On the Morawetz-Keel-Smith-Sogge inequality for the wave equation on a curved background, Publ. Res. Inst. Math. Sci., 42 (2006), 705-720.

[3]

L. Andersson and P. Blue, Hidden symmetries and decay for the wave equation on the Kerr spacetime, Ann. of Math. (2) 182 (2015), 787-853,

[4]

P. Blue and A. Soffer, Errata for "Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds", "Semilinear wave equations on the Schwarzschild manifold Ⅰ: Local Decay Estimates", and "The wave equation on the Schwarzschild metric Ⅱ: Local decay for the spin 2 Regge Wheeler equation", Preprint. ArXiv: gr-qc/0608073.

[5]

P. Blue and A. Soffer, Semilinear wave equations on the Schwarzschild manifold. Ⅰ. Local decay estimates, Adv. Differential Equations, 8 (2003), 595-614.

[6]

J. -F. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 35 (2010), 23-67,

[7]

R. Booth, H. Christianson, J. Metcalfe and J. Perry, Localized energy for wave equations with degenerate trapping, In preparation.

[8]

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), 1675-1683,

[9]

M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes Ⅰ-Ⅱ: The cases $|a|\ll M$ or axisymmetry, Preprint. ArXiv: 1010.5132.

[10]

M. Dafermos and I. Rodnianski, A note on energy currents and decay for the wave equation on a Schwarzschild background, Preprint. ArXiv: 0710.0171.

[11]

M. Dafermos and I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math. 62 (2009), 859-919,

[12]

M. Dafermos and I. Rodnianski, A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math. 185 (2011), 467-559,

[13]

M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes Ⅲ: The full subextremal case |a| < M, Ann. of Math. (2) 183 (2016), 787-913,

[14]

D. Del Santo, Global existence and blow-up for a hyperbolic system in three space dimensions, Rend. Istit. Mat. Univ. Trieste, 29 (1997), 115-140 (1998).

[15]

D. Del Santo and È. Mitidieri, Blow-up of solutions of a hyperbolic system: the critical case, Differ. Uravn., 34 (1998), 1155-1161,1293.

[16]

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), vol. 32 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1997,117-140.

[17]

K. Deng, Nonexistence of global solutions of a nonlinear hyperbolic system, Trans. Amer. Math. Soc. 349 (1997), 1685-1696,

[18]

Y. Du, J. Metcalfe, C. D. Sogge and Y. Zhou, Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations 33 (2008), 1487-1506,

[19]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math. 23 (2011), 181-205,

[20]

V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math. 119 (1997), 1291-1319,

[21]

V. Georgiev, H. Takamura and Z. Yi, The lifespan of solutions to nonlinear systems of a high-dimensional wave equation, Nonlinear Anal. 64 (2006), 2215-2250,

[22]

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,

[23]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268,

[24]

S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N. M., 1984), vol. 23 of Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 1986,293-326.

[25]

H. Kubo and M. Ohta, Critical blowup for systems of semilinear wave equations in low space dimensions, J. Math. Anal. Appl. 240 (1999), 340-360,

[26]

H. Kubo and M. Ohta, On the global behavior of classical solutions to coupled systems of semilinear wave equations, in New trends in the theory of hyperbolic equations vol. 159 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 2005,113-211,

[27]

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,

[28]

J. Marzuola, J. Metcalfe, D. Tataru and M. Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), 37-83,

[29]

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,

[30]

J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 (2007), 521-549,

[31]

J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Preprint. ArXiv: 1703.08064.

[32]

J. Metcalfe and D. Tataru, Decay estimates for variable coefficient wave equations in exterior domains, in Advances in Phase Space Analysis of Partial Differential Equations vol. 78 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Inc., Boston, MA, 2009,201-216,

[33]

J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann. 353 (2012), 1183-1237,

[34]

J. MetcalfeD. Tataru and M. Tohaneanu, Price's law on nonstationary space-times, Adv. Math., 230 (2012), 995-1028.

[35]

J. Metcalfe and C. Wang, The Strauss conjecture on asymptotically flat space-times, Comm. Pure Appl. Anal., to appear. ArXiv: 1605.02157.

[36]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A, 306 (1968), 291-296.

[37]

J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.

[38]

J. Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes, Anal. PDE, 8 (2015), 1379-1420.

[39]

J. Schaeffer, The equation $u_{tt}-Δ u = \vert u\vert ^p $ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31-44.

[40]

T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.

[41]

H. F. SmithC. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-wave equations in two dimensions with applications, Trans. Amer. Math. Soc., 364 (2012), 3329-3347.

[42]

C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.

[43]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. , 187-231. With an appendix by Igor Rodnianski.

[44]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.

[45]

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).

[46]

D. Tataru, Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math., 135 (2013), 361-401.

[47]

D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN, (), 248-292.

[48]

C. Wang, Long time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174.

[49]

C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.

[50]

C. Wang and X. Yu, Recent works on the Strauss conjecture, in Recent advances in harmonic analysis and partial differential equations, vol. 581 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2012,235-256.

[51]

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.

[52]

X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential Integral Equations, 24 (2011), 443-468.

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,

[2]

S. Alinhac, On the Morawetz-Keel-Smith-Sogge inequality for the wave equation on a curved background, Publ. Res. Inst. Math. Sci., 42 (2006), 705-720.

[3]

L. Andersson and P. Blue, Hidden symmetries and decay for the wave equation on the Kerr spacetime, Ann. of Math. (2) 182 (2015), 787-853,

[4]

P. Blue and A. Soffer, Errata for "Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds", "Semilinear wave equations on the Schwarzschild manifold Ⅰ: Local Decay Estimates", and "The wave equation on the Schwarzschild metric Ⅱ: Local decay for the spin 2 Regge Wheeler equation", Preprint. ArXiv: gr-qc/0608073.

[5]

P. Blue and A. Soffer, Semilinear wave equations on the Schwarzschild manifold. Ⅰ. Local decay estimates, Adv. Differential Equations, 8 (2003), 595-614.

[6]

J. -F. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 35 (2010), 23-67,

[7]

R. Booth, H. Christianson, J. Metcalfe and J. Perry, Localized energy for wave equations with degenerate trapping, In preparation.

[8]

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), 1675-1683,

[9]

M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes Ⅰ-Ⅱ: The cases $|a|\ll M$ or axisymmetry, Preprint. ArXiv: 1010.5132.

[10]

M. Dafermos and I. Rodnianski, A note on energy currents and decay for the wave equation on a Schwarzschild background, Preprint. ArXiv: 0710.0171.

[11]

M. Dafermos and I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math. 62 (2009), 859-919,

[12]

M. Dafermos and I. Rodnianski, A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math. 185 (2011), 467-559,

[13]

M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes Ⅲ: The full subextremal case |a| < M, Ann. of Math. (2) 183 (2016), 787-913,

[14]

D. Del Santo, Global existence and blow-up for a hyperbolic system in three space dimensions, Rend. Istit. Mat. Univ. Trieste, 29 (1997), 115-140 (1998).

[15]

D. Del Santo and È. Mitidieri, Blow-up of solutions of a hyperbolic system: the critical case, Differ. Uravn., 34 (1998), 1155-1161,1293.

[16]

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), vol. 32 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1997,117-140.

[17]

K. Deng, Nonexistence of global solutions of a nonlinear hyperbolic system, Trans. Amer. Math. Soc. 349 (1997), 1685-1696,

[18]

Y. Du, J. Metcalfe, C. D. Sogge and Y. Zhou, Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions, Comm. Partial Differential Equations 33 (2008), 1487-1506,

[19]

D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math. 23 (2011), 181-205,

[20]

V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math. 119 (1997), 1291-1319,

[21]

V. Georgiev, H. Takamura and Z. Yi, The lifespan of solutions to nonlinear systems of a high-dimensional wave equation, Nonlinear Anal. 64 (2006), 2215-2250,

[22]

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,

[23]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268,

[24]

S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N. M., 1984), vol. 23 of Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 1986,293-326.

[25]

H. Kubo and M. Ohta, Critical blowup for systems of semilinear wave equations in low space dimensions, J. Math. Anal. Appl. 240 (1999), 340-360,

[26]

H. Kubo and M. Ohta, On the global behavior of classical solutions to coupled systems of semilinear wave equations, in New trends in the theory of hyperbolic equations vol. 159 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 2005,113-211,

[27]

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,

[28]

J. Marzuola, J. Metcalfe, D. Tataru and M. Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), 37-83,

[29]

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,

[30]

J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 (2007), 521-549,

[31]

J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Preprint. ArXiv: 1703.08064.

[32]

J. Metcalfe and D. Tataru, Decay estimates for variable coefficient wave equations in exterior domains, in Advances in Phase Space Analysis of Partial Differential Equations vol. 78 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Inc., Boston, MA, 2009,201-216,

[33]

J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann. 353 (2012), 1183-1237,

[34]

J. MetcalfeD. Tataru and M. Tohaneanu, Price's law on nonstationary space-times, Adv. Math., 230 (2012), 995-1028.

[35]

J. Metcalfe and C. Wang, The Strauss conjecture on asymptotically flat space-times, Comm. Pure Appl. Anal., to appear. ArXiv: 1605.02157.

[36]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A, 306 (1968), 291-296.

[37]

J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.

[38]

J. Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes, Anal. PDE, 8 (2015), 1379-1420.

[39]

J. Schaeffer, The equation $u_{tt}-Δ u = \vert u\vert ^p $ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31-44.

[40]

T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.

[41]

H. F. SmithC. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-wave equations in two dimensions with applications, Trans. Amer. Math. Soc., 364 (2012), 3329-3347.

[42]

C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.

[43]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. , 187-231. With an appendix by Igor Rodnianski.

[44]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.

[45]

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).

[46]

D. Tataru, Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math., 135 (2013), 361-401.

[47]

D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN, (), 248-292.

[48]

C. Wang, Long time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174.

[49]

C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.

[50]

C. Wang and X. Yu, Recent works on the Strauss conjecture, in Recent advances in harmonic analysis and partial differential equations, vol. 581 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2012,235-256.

[51]

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.

[52]

X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential Integral Equations, 24 (2011), 443-468.

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