# American Institute of Mathematical Sciences

May  2017, 16(3): 719-744. doi: 10.3934/cpaa.2017035

## Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

 1 School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical, Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, 210023, China 2 Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, Göttingen, D-37073, Germany

* Corresponding author

Received  January 2016 Revised  March 2016 Published  February 2016

Fund Project: The first author and the third author were supported by the NSFC (No. 11571177) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The second author was supported by the DFG via the Sino-German project "Analysis of PDEs and application."

For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in [1,2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

Citation: 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
##### References:
 [1] S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), 97-127. doi: 10.2307/121020. Google Scholar [2] S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23. doi: 10.1007/BF02392822. Google Scholar [3] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. Ⅱ, Amer. J. Math., 123 (2001), 1071-1101. Google Scholar [4] S. Alinhac, An example of blowup at infinity for quasilinear wave equations, Asterisque, 284 (2003), 1-91. Google Scholar [5] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205. Google Scholar [6] D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014. Google Scholar [7] Binbing Ding, Yingbo Liu and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅰ, SIAM Journal on Mathematical Analysis, 47 (2015), 4192-4228. doi: 10.1137/151004793. Google Scholar [8] Binbing Ding, Ingo Witt and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅱ, J. Differential Equations, 261 (2016), 1429-1471. doi: 10.1016/j.jde.2016.04.002. Google Scholar [9] Binbing Ding, Ingo Witt and Huicheng Yin, Blowup of classical solutions for 2-D quasilinear wave equations with small initial data, Quart. Appl. Math., 73 (2015), 773-796. doi: 10.1090/qam/1410. Google Scholar [10] Binbing Ding, Ingo Witt and Huicheng Yin, On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations, 252 (2012), 3608-3629. doi: 10.1016/j.jde.2011.11.018. Google Scholar [11] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955. Google Scholar [12] Fei Guo and phenomena Wave-breaking, decay properties and limit behaviour of solutions of the Degasperis-Procesi equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 805-824. doi: 10.1017/S0308210511000321. Google Scholar [13] Fei Guo and Weiwei Peng, Blowup solutions for the generalized two-component CamassaHolm system on the circle, Nonlinear Anal., 105 (2014), 120-133. doi: 10.1016/j.na.2014.03.021. Google Scholar [14] L. Hömander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, MittagLeffler report No. 5,1985.Google Scholar [15] L. Hömander, Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26, Springer Verlag, Heidelberg, 1997.Google Scholar [16] A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J., 24 (1995), 575-615. doi: 10.14492/hokmj/1380892610. Google Scholar [17] F. John, Blow-up of radial solutions of utt = c2(ut)∆u in three space dimensions, Mat. Apl. Comput., 4 (1985), 3-18. Google Scholar [18] F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403. Google Scholar [19] M. Keel, H. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc., 17 (2004), 109-153. doi: 10.1090/S0894-0347-03-00443-0. Google Scholar [20] S. Klainerman, The Null Condition and Global Existence to Nonlinear Wave Equations, Lectures in Appl. Math. , 23, Amer. Math. Soc. , Providence, RI, 1986. Google Scholar [21] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ℝn+1, Comm. Pure Appl. Math., 40 (1987), 111-117. doi: 10.1002/cpa.3160400105. Google Scholar [22] Yutian Lei, Singularity analysis of Ginzburg-Landau energy related to p-wave superconductivity, Z. Angew. Math. Phys., 64 (2013), 1249-1266. doi: 10.1007/s00033-012-0284-y. Google Scholar [23] Ta-tsien Li and Yun-mei Chen, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations, 13 (1988), 383-422. doi: 10.1080/03605308808820547. Google Scholar [24] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. doi: 10.1002/cpa.3160430403. Google Scholar [25] H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. doi: 10.1002/cpa.3160450902. Google Scholar [26] H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157. doi: 10.1353/ajm.2008.0009. Google Scholar [27] H. Lindblad, M. Nakamura and C. D. Sogge, Remarks on global solutions for nonlinear wave equations under the standard null conditions, J. Differential Equations, 254 (2013), 1396-1436. doi: 10.1016/j.jde.2012.10.022. Google Scholar [28] J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, preprint, arXiv: 1407.6320.Google Scholar [29] Sijue Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220. doi: 10.1007/s00222-010-0288-1. Google Scholar [30] Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8. Google Scholar

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##### References:
 [1] S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, Ann. of Math., 149 (1999), 97-127. doi: 10.2307/121020. Google Scholar [2] S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. Ⅱ, Acta Math., 182 (1999), 1-23. doi: 10.1007/BF02392822. Google Scholar [3] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. Ⅱ, Amer. J. Math., 123 (2001), 1071-1101. Google Scholar [4] S. Alinhac, An example of blowup at infinity for quasilinear wave equations, Asterisque, 284 (2003), 1-91. Google Scholar [5] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205. Google Scholar [6] D. Christodoulou and S. Miao, Compressible Flow and Euler's Equations, Surveys of Modern Mathematics, 9. International Press, Somerville, MA; Higher Education Press, Beijing, 2014. Google Scholar [7] Binbing Ding, Yingbo Liu and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅰ, SIAM Journal on Mathematical Analysis, 47 (2015), 4192-4228. doi: 10.1137/151004793. Google Scholar [8] Binbing Ding, Ingo Witt and Huicheng Yin, The small data solutions of general 3-D quasilinear wave equations. Ⅱ, J. Differential Equations, 261 (2016), 1429-1471. doi: 10.1016/j.jde.2016.04.002. Google Scholar [9] Binbing Ding, Ingo Witt and Huicheng Yin, Blowup of classical solutions for 2-D quasilinear wave equations with small initial data, Quart. Appl. Math., 73 (2015), 773-796. doi: 10.1090/qam/1410. Google Scholar [10] Binbing Ding, Ingo Witt and Huicheng Yin, On the blowup of classical solutions to the 3-D pressure-gradient systems, J. Differential Equations, 252 (2012), 3608-3629. doi: 10.1016/j.jde.2011.11.018. Google Scholar [11] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955. Google Scholar [12] Fei Guo and phenomena Wave-breaking, decay properties and limit behaviour of solutions of the Degasperis-Procesi equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 805-824. doi: 10.1017/S0308210511000321. Google Scholar [13] Fei Guo and Weiwei Peng, Blowup solutions for the generalized two-component CamassaHolm system on the circle, Nonlinear Anal., 105 (2014), 120-133. doi: 10.1016/j.na.2014.03.021. Google Scholar [14] L. Hömander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, MittagLeffler report No. 5,1985.Google Scholar [15] L. Hömander, Lectures on Nonlinear Hyperbolic Equations, Mathematiques & Applications 26, Springer Verlag, Heidelberg, 1997.Google Scholar [16] A. Hoshiga, The asymptotic behaviour of the radially symmetric solutions to quasilinear wave equations in two space dimensions, Hokkaido Math. J., 24 (1995), 575-615. doi: 10.14492/hokmj/1380892610. Google Scholar [17] F. John, Blow-up of radial solutions of utt = c2(ut)∆u in three space dimensions, Mat. Apl. Comput., 4 (1985), 3-18. Google Scholar [18] F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403. Google Scholar [19] M. Keel, H. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc., 17 (2004), 109-153. doi: 10.1090/S0894-0347-03-00443-0. Google Scholar [20] S. Klainerman, The Null Condition and Global Existence to Nonlinear Wave Equations, Lectures in Appl. Math. , 23, Amer. Math. Soc. , Providence, RI, 1986. Google Scholar [21] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ℝn+1, Comm. Pure Appl. Math., 40 (1987), 111-117. doi: 10.1002/cpa.3160400105. Google Scholar [22] Yutian Lei, Singularity analysis of Ginzburg-Landau energy related to p-wave superconductivity, Z. Angew. Math. Phys., 64 (2013), 1249-1266. doi: 10.1007/s00033-012-0284-y. Google Scholar [23] Ta-tsien Li and Yun-mei Chen, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations, 13 (1988), 383-422. doi: 10.1080/03605308808820547. Google Scholar [24] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. doi: 10.1002/cpa.3160430403. Google Scholar [25] H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. doi: 10.1002/cpa.3160450902. Google Scholar [26] H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157. doi: 10.1353/ajm.2008.0009. Google Scholar [27] H. Lindblad, M. Nakamura and C. D. Sogge, Remarks on global solutions for nonlinear wave equations under the standard null conditions, J. Differential Equations, 254 (2013), 1396-1436. doi: 10.1016/j.jde.2012.10.022. Google Scholar [28] J. Speck, Shock formation in small-data solutions to 3D quasilinear wave equations, preprint, arXiv: 1407.6320.Google Scholar [29] Sijue Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220. doi: 10.1007/s00222-010-0288-1. Google Scholar [30] Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135. doi: 10.1007/s00222-009-0176-8. Google Scholar
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