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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,  Google Scholar [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.   Google Scholar [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,  Google Scholar [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.  Google Scholar [5] P. Blue and A. Soffer, Semilinear wave equations on the Schwarzschild manifold. Ⅰ. Local decay estimates, Adv. Differential Equations, 8 (2003), 595-614.   Google Scholar [6] J. -F. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 35 (2010), 23-67,  Google Scholar [7] R. Booth, H. Christianson, J. Metcalfe and J. Perry, Localized energy for wave equations with degenerate trapping, In preparation. Google Scholar [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,  Google Scholar [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. Google Scholar [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. Google Scholar [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,  Google Scholar [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,  Google Scholar [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,  Google Scholar [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).   Google Scholar [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.   Google Scholar [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.  Google Scholar [17] K. Deng, Nonexistence of global solutions of a nonlinear hyperbolic system, Trans. Amer. Math. Soc. 349 (1997), 1685-1696,  Google Scholar [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,  Google Scholar [19] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math. 23 (2011), 181-205,  Google Scholar [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,  Google Scholar [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,  Google Scholar [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,  Google Scholar [23] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268,  Google Scholar [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.  Google Scholar [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,  Google Scholar [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,  Google Scholar [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,  Google Scholar [28] J. Marzuola, J. Metcalfe, D. Tataru and M. Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), 37-83,  Google Scholar [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,  Google Scholar [30] J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 (2007), 521-549,  Google Scholar [31] J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Preprint. ArXiv: 1703.08064. Google Scholar [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,  Google Scholar [33] J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann. 353 (2012), 1183-1237,  Google Scholar [34] J. Metcalfe, D. Tataru and M. Tohaneanu, Price's law on nonstationary space-times, Adv. Math., 230 (2012), 995-1028.   Google Scholar [35] J. Metcalfe and C. Wang, The Strauss conjecture on asymptotically flat space-times, Comm. Pure Appl. Anal., to appear. ArXiv: 1605.02157. Google Scholar [36] C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A, 306 (1968), 291-296.   Google Scholar [37] J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.   Google Scholar [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.   Google Scholar [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.   Google Scholar [40] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.   Google Scholar [41] H. F. Smith, C. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-wave equations in two dimensions with applications, Trans. Amer. Math. Soc., 364 (2012), 3329-3347.   Google Scholar [42] C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.   Google Scholar [43] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. , 187-231. With an appendix by Igor Rodnianski.  Google Scholar [44] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.   Google Scholar [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).   Google Scholar [46] D. Tataru, Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math., 135 (2013), 361-401.   Google Scholar [47] D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN, (), 248-292.   Google Scholar [48] C. Wang, Long time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174.   Google Scholar [49] C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.   Google Scholar [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.  Google Scholar [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.   Google Scholar [52] X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential Integral Equations, 24 (2011), 443-468.   Google Scholar

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,  Google Scholar [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.   Google Scholar [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,  Google Scholar [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.  Google Scholar [5] P. Blue and A. Soffer, Semilinear wave equations on the Schwarzschild manifold. Ⅰ. Local decay estimates, Adv. Differential Equations, 8 (2003), 595-614.   Google Scholar [6] J. -F. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations 35 (2010), 23-67,  Google Scholar [7] R. Booth, H. Christianson, J. Metcalfe and J. Perry, Localized energy for wave equations with degenerate trapping, In preparation. Google Scholar [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,  Google Scholar [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. Google Scholar [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. Google Scholar [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,  Google Scholar [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,  Google Scholar [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,  Google Scholar [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).   Google Scholar [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.   Google Scholar [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.  Google Scholar [17] K. Deng, Nonexistence of global solutions of a nonlinear hyperbolic system, Trans. Amer. Math. Soc. 349 (1997), 1685-1696,  Google Scholar [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,  Google Scholar [19] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math. 23 (2011), 181-205,  Google Scholar [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,  Google Scholar [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,  Google Scholar [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,  Google Scholar [23] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268,  Google Scholar [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.  Google Scholar [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,  Google Scholar [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,  Google Scholar [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,  Google Scholar [28] J. Marzuola, J. Metcalfe, D. Tataru and M. Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), 37-83,  Google Scholar [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,  Google Scholar [30] J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 (2007), 521-549,  Google Scholar [31] J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Preprint. ArXiv: 1703.08064. Google Scholar [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,  Google Scholar [33] J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann. 353 (2012), 1183-1237,  Google Scholar [34] J. Metcalfe, D. Tataru and M. Tohaneanu, Price's law on nonstationary space-times, Adv. Math., 230 (2012), 995-1028.   Google Scholar [35] J. Metcalfe and C. Wang, The Strauss conjecture on asymptotically flat space-times, Comm. Pure Appl. Anal., to appear. ArXiv: 1605.02157. Google Scholar [36] C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A, 306 (1968), 291-296.   Google Scholar [37] J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.   Google Scholar [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.   Google Scholar [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.   Google Scholar [40] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.   Google Scholar [41] H. F. Smith, C. D. Sogge and C. Wang, Strichartz estimates for Dirichlet-wave equations in two dimensions with applications, Trans. Amer. Math. Soc., 364 (2012), 3329-3347.   Google Scholar [42] C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32.   Google Scholar [43] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, Int. Math. Res. Not. , 187-231. With an appendix by Igor Rodnianski.  Google Scholar [44] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.   Google Scholar [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).   Google Scholar [46] D. Tataru, Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math., 135 (2013), 361-401.   Google Scholar [47] D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN, (), 248-292.   Google Scholar [48] C. Wang, Long time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174.   Google Scholar [49] C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566.   Google Scholar [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.  Google Scholar [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.   Google Scholar [52] X. Yu, Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture, Differential Integral Equations, 24 (2011), 443-468.   Google Scholar
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