The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions

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May 2014, 13(3): 1167-1186. doi: 10.3934/cpaa.2014.13.1167

The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions

Dongbing Zha ^1, and Yi Zhou ^1,

1.	School of Mathematical Sciences, Fudan University, Shanghai, 200433, China, China

Received April 2013 Revised September 2013 Published December 2013

Abstract
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We study the Cauchy problem for systems of quasilinear wave equations with multiple propagation speeds in four space dimensions. The nonlinear term in this problem may explicitly depends on the unknown function itself. By some new $L^{\infty}_{t}L^2_{x}$ estimates of the unknown function, combining with some Klainerman--Sideris type weighted estimates, we get the sharp lower bound of lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$ for the quasilinear system.

Keywords: multiple speeds, Quasilinear wave equations, lifespan..

Mathematics Subject Classification: 35L05, 35L10, 35L7.

Citation: Dongbing Zha, Yi Zhou. The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1167-1186. doi: 10.3934/cpaa.2014.13.1167

References:

[1]	Y. Du and Z. A. Yao, The lifespan for quasi-linear wave equations with multiple-speeds in space dimensions $n\geq3$ , Manuscripta Math., 132 (2010), 343-364. doi: 10.1007/s00229-010-0350-8.
[2]	Y. Du and Y. Zhou, The lifespan for nonlinear wave equations with multiple propagations speeds in 3D , Nonlinear Anal., 68 (2008), 1793-1801. doi: 10.1016/j.na.2007.01.012.
[3]	K. Hidano, An elementary proof of global or almost global existence for quasilinear wave equations , Tohoku Math. J., 56 (2004), 271-287. doi: 10.2748/tmj/1113246554.
[4]	K. Hidano, J. Metcalfe, H. 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.
[5]	L. Hörmander, On the fully nonlinear cauchy problem with small data II , in Microlocal Analysis and Nonlinear Waves, IMA Vol. Math. Appl., 30 (eds. Michael Beals, Richard B. Melrose and Jeffrey Rauch), Springer-Verlag, (1991), 51-81. doi: 10.1007/978-1-4613-9136-4_6.
[6]	S. Katayama, Global and almost global existence for systems of nonlinear wave equations with different propagation speeds , Differential Integral Equations, 17 (2004), 1043-1078.
[7]	S. Katayama, Global existence for a class of systems of nonlinear wave equations in three dimensions , Chinese Ann. Math. B, 25 (2004), 463-482. doi: 10.1142/S0252959904000421.
[8]	S. Katayama, Global existence for systems of wave equations with nonresonant nonlinearities and null forms , J. Differential Equations, 209 (2005), 140-171. doi: 10.1016/j.jde.2004.09.012.
[9]	S. Klainerman, Uniform decay estimate and the Lorentz invariance of the classical wave equations , Comm. Pure Appl. Math., 38 (1985), 321-332.
[10]	S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D , Comm. Pure Appl. Math., 49 (1996), 307-321. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H.
[11]	T. T. Li and Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations, Longman Scientific & Technical UK, 1992.
[12]	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.
[13]	H. Lindblad and C. D. Sogge, Long time existence for small amplitude semilinear wave equation , Amer. J. Math., 118 (1996), 1047-1135. doi: 10.1353/ajm.1996.0042.
[14]	J.Metcalfe and C. D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles , Discrete Contin. Dyn. Syst., 28 (2010), 1589-1601. doi: 10.3934/dcds.2010.28.1589.
[15]	C. X. Miao, J. H. Wu and Z. F. Zhang, Littlewood-Paley Theory and Its Application in Hydrodynamic Equations (Chinese Edition), Monographs on Modern Pure Mathematics 142, Beijing, Science Press, 2012.
[16]	T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds , SIAM J. Math. Anal., 33 (2001), 477-488. doi: 10.1137/S0036141000378966.
[17]	H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions , J. Differential Equations, 251 (2011), 1157-1171. doi: 10.1016/j.jde.2011.03.024.
[18]	C. B. Wang and X. Yu, Recent works on the Strauss conjecture , in Recent Advances in Harmonic Analysis and Partial Differential Equations, Contemp. Math. 581 (eds. Andrea R. Nahmod et al.), Amer. Math. Soc., Providence, RI, (2012), 235-256. doi: 10.1090/conm/581/11497.

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References:

[1]	Y. Du and Z. A. Yao, The lifespan for quasi-linear wave equations with multiple-speeds in space dimensions $n\geq3$ , Manuscripta Math., 132 (2010), 343-364. doi: 10.1007/s00229-010-0350-8.
[2]	Y. Du and Y. Zhou, The lifespan for nonlinear wave equations with multiple propagations speeds in 3D , Nonlinear Anal., 68 (2008), 1793-1801. doi: 10.1016/j.na.2007.01.012.
[3]	K. Hidano, An elementary proof of global or almost global existence for quasilinear wave equations , Tohoku Math. J., 56 (2004), 271-287. doi: 10.2748/tmj/1113246554.
[4]	K. Hidano, J. Metcalfe, H. 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.
[5]	L. Hörmander, On the fully nonlinear cauchy problem with small data II , in Microlocal Analysis and Nonlinear Waves, IMA Vol. Math. Appl., 30 (eds. Michael Beals, Richard B. Melrose and Jeffrey Rauch), Springer-Verlag, (1991), 51-81. doi: 10.1007/978-1-4613-9136-4_6.
[6]	S. Katayama, Global and almost global existence for systems of nonlinear wave equations with different propagation speeds , Differential Integral Equations, 17 (2004), 1043-1078.
[7]	S. Katayama, Global existence for a class of systems of nonlinear wave equations in three dimensions , Chinese Ann. Math. B, 25 (2004), 463-482. doi: 10.1142/S0252959904000421.
[8]	S. Katayama, Global existence for systems of wave equations with nonresonant nonlinearities and null forms , J. Differential Equations, 209 (2005), 140-171. doi: 10.1016/j.jde.2004.09.012.
[9]	S. Klainerman, Uniform decay estimate and the Lorentz invariance of the classical wave equations , Comm. Pure Appl. Math., 38 (1985), 321-332.
[10]	S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D , Comm. Pure Appl. Math., 49 (1996), 307-321. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H.
[11]	T. T. Li and Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations, Longman Scientific & Technical UK, 1992.
[12]	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.
[13]	H. Lindblad and C. D. Sogge, Long time existence for small amplitude semilinear wave equation , Amer. J. Math., 118 (1996), 1047-1135. doi: 10.1353/ajm.1996.0042.
[14]	J.Metcalfe and C. D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles , Discrete Contin. Dyn. Syst., 28 (2010), 1589-1601. doi: 10.3934/dcds.2010.28.1589.
[15]	C. X. Miao, J. H. Wu and Z. F. Zhang, Littlewood-Paley Theory and Its Application in Hydrodynamic Equations (Chinese Edition), Monographs on Modern Pure Mathematics 142, Beijing, Science Press, 2012.
[16]	T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds , SIAM J. Math. Anal., 33 (2001), 477-488. doi: 10.1137/S0036141000378966.
[17]	H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions , J. Differential Equations, 251 (2011), 1157-1171. doi: 10.1016/j.jde.2011.03.024.
[18]	C. B. Wang and X. Yu, Recent works on the Strauss conjecture , in Recent Advances in Harmonic Analysis and Partial Differential Equations, Contemp. Math. 581 (eds. Andrea R. Nahmod et al.), Amer. Math. Soc., Providence, RI, (2012), 235-256. doi: 10.1090/conm/581/11497.

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