# American Institute of Mathematical Sciences

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

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

Received  April 2013 Revised  September 2013 Published  December 2013

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.
Citation: Dongbing Zha, Yi Zhou. The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions. Communications on Pure & 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 $n\geq3$ ,, \emph{Manuscripta Math.}, 132 (2010), 343. doi: 10.1007/s00229-010-0350-8. Google Scholar [2] Y. Du and Y. Zhou, The lifespan for nonlinear wave equations with multiple propagations speeds in 3D ,, \emph{Nonlinear Anal.}, 68 (2008), 1793. doi: 10.1016/j.na.2007.01.012. Google Scholar [3] K. Hidano, An elementary proof of global or almost global existence for quasilinear wave equations ,, \emph{Tohoku Math. J.}, 56 (2004), 271. doi: 10.2748/tmj/1113246554. Google Scholar [4] K. Hidano, J. Metcalfe, H. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles ,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 2789. doi: 10.1090/S0002-9947-09-05053-3. Google Scholar [5] L. Hörmander, On the fully nonlinear cauchy problem with small data II ,, in \emph{Microlocal Analysis and Nonlinear Waves}, (1991), 51. doi: 10.1007/978-1-4613-9136-4_6. Google Scholar [6] S. Katayama, Global and almost global existence for systems of nonlinear wave equations with different propagation speeds ,, \emph{Differential Integral Equations}, 17 (2004), 1043. Google Scholar [7] S. Katayama, Global existence for a class of systems of nonlinear wave equations in three dimensions ,, \emph{Chinese Ann. Math. B}, 25 (2004), 463. doi: 10.1142/S0252959904000421. Google Scholar [8] S. Katayama, Global existence for systems of wave equations with nonresonant nonlinearities and null forms ,, \emph{J. Differential Equations}, 209 (2005), 140. doi: 10.1016/j.jde.2004.09.012. Google Scholar [9] S. Klainerman, Uniform decay estimate and the Lorentz invariance of the classical wave equations ,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 321. Google Scholar [10] S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D ,, \emph{Comm. Pure Appl. Math.}, 49 (1996), 307. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H. Google Scholar [11] T. T. Li and Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations,, Longman Scientific & Technical UK, (1992). Google Scholar [12] T. T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions ,, \emph{Indiana Univ. Math. J.}, 44 (1995), 1207. doi: 10.1512/iumj.1995.44.2026. Google Scholar [13] H. Lindblad and C. D. Sogge, Long time existence for small amplitude semilinear wave equation ,, \emph{Amer. J. Math.}, 118 (1996), 1047. doi: 10.1353/ajm.1996.0042. Google Scholar [14] J.Metcalfe and C. D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles ,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1589. doi: 10.3934/dcds.2010.28.1589. Google Scholar [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, (2012). Google Scholar [16] T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds ,, \emph{SIAM J. Math. Anal.}, 33 (2001), 477. doi: 10.1137/S0036141000378966. Google Scholar [17] H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions ,, \emph{J. Differential Equations}, 251 (2011), 1157. doi: 10.1016/j.jde.2011.03.024. Google Scholar [18] C. B. Wang and X. Yu, Recent works on the Strauss conjecture ,, in \emph{Recent Advances in Harmonic Analysis and Partial Differential Equations}, (2012), 235. doi: 10.1090/conm/581/11497. Google Scholar

show all references

##### References:
 [1] Y. Du and Z. A. Yao, The lifespan for quasi-linear wave equations with multiple-speeds in space dimensions $n\geq3$ ,, \emph{Manuscripta Math.}, 132 (2010), 343. doi: 10.1007/s00229-010-0350-8. Google Scholar [2] Y. Du and Y. Zhou, The lifespan for nonlinear wave equations with multiple propagations speeds in 3D ,, \emph{Nonlinear Anal.}, 68 (2008), 1793. doi: 10.1016/j.na.2007.01.012. Google Scholar [3] K. Hidano, An elementary proof of global or almost global existence for quasilinear wave equations ,, \emph{Tohoku Math. J.}, 56 (2004), 271. doi: 10.2748/tmj/1113246554. Google Scholar [4] K. Hidano, J. Metcalfe, H. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles ,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 2789. doi: 10.1090/S0002-9947-09-05053-3. Google Scholar [5] L. Hörmander, On the fully nonlinear cauchy problem with small data II ,, in \emph{Microlocal Analysis and Nonlinear Waves}, (1991), 51. doi: 10.1007/978-1-4613-9136-4_6. Google Scholar [6] S. Katayama, Global and almost global existence for systems of nonlinear wave equations with different propagation speeds ,, \emph{Differential Integral Equations}, 17 (2004), 1043. Google Scholar [7] S. Katayama, Global existence for a class of systems of nonlinear wave equations in three dimensions ,, \emph{Chinese Ann. Math. B}, 25 (2004), 463. doi: 10.1142/S0252959904000421. Google Scholar [8] S. Katayama, Global existence for systems of wave equations with nonresonant nonlinearities and null forms ,, \emph{J. Differential Equations}, 209 (2005), 140. doi: 10.1016/j.jde.2004.09.012. Google Scholar [9] S. Klainerman, Uniform decay estimate and the Lorentz invariance of the classical wave equations ,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 321. Google Scholar [10] S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D ,, \emph{Comm. Pure Appl. Math.}, 49 (1996), 307. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H. Google Scholar [11] T. T. Li and Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations,, Longman Scientific & Technical UK, (1992). Google Scholar [12] T. T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions ,, \emph{Indiana Univ. Math. J.}, 44 (1995), 1207. doi: 10.1512/iumj.1995.44.2026. Google Scholar [13] H. Lindblad and C. D. Sogge, Long time existence for small amplitude semilinear wave equation ,, \emph{Amer. J. Math.}, 118 (1996), 1047. doi: 10.1353/ajm.1996.0042. Google Scholar [14] J.Metcalfe and C. D. Sogge, Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles ,, \emph{Discrete Contin. Dyn. Syst.}, 28 (2010), 1589. doi: 10.3934/dcds.2010.28.1589. Google Scholar [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, (2012). Google Scholar [16] T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds ,, \emph{SIAM J. Math. Anal.}, 33 (2001), 477. doi: 10.1137/S0036141000378966. Google Scholar [17] H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions ,, \emph{J. Differential Equations}, 251 (2011), 1157. doi: 10.1016/j.jde.2011.03.024. Google Scholar [18] C. B. Wang and X. Yu, Recent works on the Strauss conjecture ,, in \emph{Recent Advances in Harmonic Analysis and Partial Differential Equations}, (2012), 235. doi: 10.1090/conm/581/11497. Google Scholar
 [1] Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1265-1283. doi: 10.3934/cpaa.2016.15.1265 [2] Dongbing Zha. A note on quasilinear wave equations in two space dimensions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2855-2871. doi: 10.3934/dcds.2016.36.2855 [3] Fathi Dkhil, Angela Stevens. Traveling wave speeds in rapidly oscillating media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 89-108. doi: 10.3934/dcds.2009.25.89 [4] Yuusuke Sugiyama. Degeneracy in finite time of 1D quasilinear wave equations Ⅱ. Evolution Equations & Control Theory, 2017, 6 (4) : 615-628. doi: 10.3934/eect.2017031 [5] Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547 [6] Francisco Odair de Paiva. Multiple solutions for a class of quasilinear problems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 669-680. doi: 10.3934/dcds.2006.15.669 [7] Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713 [8] Linghai Zhang, Ping-Shi Wu, Melissa Anne Stoner. Influence of neurobiological mechanisms on speeds of traveling wave fronts in mathematical neuroscience. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 1003-1037. doi: 10.3934/dcdsb.2011.16.1003 [9] Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663 [10] Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589 [11] Bingbing Ding, Ingo Witt, Huicheng Yin. Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 719-744. doi: 10.3934/cpaa.2017035 [12] Dongsheng Kang. Quasilinear systems involving multiple critical exponents and potentials. Communications on Pure & Applied Analysis, 2013, 12 (2) : 695-710. doi: 10.3934/cpaa.2013.12.695 [13] Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128 [14] Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure & Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046 [15] Yuji Sagawa, Hideaki Sunagawa. The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5743-5761. doi: 10.3934/dcds.2016052 [16] Feng Cao, Wenxian Shen. Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4697-4727. doi: 10.3934/dcds.2017202 [17] Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069 [18] Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 [19] Bernard Dacorogna, Olivier Kneuss. Multiple Jacobian equations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1779-1787. doi: 10.3934/cpaa.2014.13.1779 [20] Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

2018 Impact Factor: 0.925