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December  2017, 6(4): 615-628. doi: 10.3934/eect.2017031

## Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

 Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan

Received  January 2017 Revised  August 2017 Published  September 2017

Fund Project: The first author is supported by Grant-in-Aid for Young Scientists Research (B), 16K17631

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2 _x u + λ c(u)c'(u)(\partial_x u)^2$ with the parameter $λ ∈ [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(· )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition so that the equation with $0≤q λ < 2$ degenerates in finite time.

Citation: 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
##### References:
 [1] W. F. Ames and R. J. Lohner, Group properties of $u_{tt}=[f(u)u_x]_x$, Int. J. Non-linear Mech., 16 (1981), 439-447. Google Scholar [2] A. Bressan and Y. Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys., 266 (2006), 471-497. doi: 10.1007/s00220-006-0047-8. Google Scholar [3] G. Chen and Y. Shen, Existence and regularity of solutions in nonlinear wave equations, Discrete Contin. Dyn. Syst. series A, 35 (2015), 3327-3342. doi: 10.3934/dcds.2015.35.3327. Google Scholar [4] R. T. Glassey, J. K. Hunter and Y. Zheng, Singularities of a variational wave equation, J. Differential Equations, 129 (1996), 49-78. doi: 10.1006/jdeq.1996.0111. Google Scholar [5] R. T. Glassey, J. K. Hunter and Y. Zheng, Singularities and oscillations in a nonlinear wave variational wave equation, Singularities and oscillations, The IMA Volumes in Mathematics and its applications, 91 (1997), 37-60. doi: 10.1007/978-1-4612-1972-9_3. Google Scholar [6] T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal., 63 (1977), 273-294. doi: 10.1007/BF00251584. Google Scholar [7] J. L. Johnson, Global continuous solutions of hyperbolic systems of quasilinear equations, Bull. Amer. Math. Soc., 73 (1967), 639-641. doi: 10.1090/S0002-9904-1967-11805-6. Google Scholar [8] K. Kato and Y. Sugiyama, Blow up of solutions to second sound equation in one space dimension, Kyushu J. Math., 67 (2013), 129-142. doi: 10.2206/kyushujm.67.129. Google Scholar [9] S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math., 33 (1980), 241-263. doi: 10.1002/cpa.3160330304. Google Scholar [10] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, Vol. 6, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959. Google Scholar [11] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. doi: 10.1063/1.1704154. Google Scholar [12] H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157. doi: 10.1353/ajm.2008.0009. Google Scholar [13] R. C. MacCamy and V. J. Mizel, Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Ration. Mech. Anal., 25 (1967), 299-320. doi: 10.1007/BF00250932. Google Scholar [14] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, Appl. Math. Sci., 1984. doi: 10.1007/978-1-4612-1116-7. Google Scholar [15] R. Manfrin, A note on the formation of singularities for quasi-linear hyperbolic systems, SIAM J. Math. Anal., 32 (2000), 261-290. Google Scholar [16] M.-Y. Qi, On the Cauchy problem for a class of hyperbolic equations with initial data on the parabolic degenerating line, Acta Math. Sinica, 8 (1958), 521-529. Google Scholar [17] Y. Sugiyama, Global existence of solutions to some quasilinear wave equation in one space dimension, Differential Integral Equations, 26 (2013), 487-504. Google Scholar [18] Y. Sugiyama, Degeneracy in finite time of 1D quasilinear wave equations, SIAM J. Math. Anal., 48 (2016), 847-860. doi: 10.1137/15M1016369. Google Scholar [19] Y. Sugiyama, Large time behavior of solutions to 1D quasilinear wave equations, RIMS Kôkyûroku Bessatsu, B63 (2017), 113-123.Google Scholar [20] K. Taniguchi and Y. Tozaki, A hyperbolic equation with double characteristics which has a solution with branching singularities, Math. Japonica, 25 (1980), 279-300. Google Scholar [21] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser Boston, 1991. doi: 10.1007/978-1-4612-0431-2. Google Scholar [22] M. K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics, Micro-Local Approach, Akademie Verlag, 1997. Google Scholar [23] M. Yamaguchi and T. Nishida, On some global solution for quasilinear hyperbolic equations, Funkcial. Ekvac., 11 (1968), 51-57. Google Scholar [24] N. J. Zabusky, Exact solution for the vibrations of a nonlinear continuous model string, J. Math. Phys., 3 (1962), 1028-1039. doi: 10.1063/1.1724290. Google Scholar [25] P. Zhang and Y. Zheng, Rarefactive solutions to a nonlinear variational wave equation of liquid crystals, Comm. Partial Differential Equations, 26 (2001), 381-419. doi: 10.1081/PDE-100002240. Google Scholar [26] P. Zhang and Y. Zheng, Singular and rarefactive solutions to a nonlinear variational wave equation, Chinese Ann. Math. Series B, 22 (2001), 159-170. doi: 10.1142/S0252959901000152. Google Scholar [27] P. Zhang and Y. Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), 303-319. doi: 10.1007/s00205-002-0232-7. Google Scholar

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##### References:
 [1] W. F. Ames and R. J. Lohner, Group properties of $u_{tt}=[f(u)u_x]_x$, Int. J. Non-linear Mech., 16 (1981), 439-447. Google Scholar [2] A. Bressan and Y. Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys., 266 (2006), 471-497. doi: 10.1007/s00220-006-0047-8. Google Scholar [3] G. Chen and Y. Shen, Existence and regularity of solutions in nonlinear wave equations, Discrete Contin. Dyn. Syst. series A, 35 (2015), 3327-3342. doi: 10.3934/dcds.2015.35.3327. Google Scholar [4] R. T. Glassey, J. K. Hunter and Y. Zheng, Singularities of a variational wave equation, J. Differential Equations, 129 (1996), 49-78. doi: 10.1006/jdeq.1996.0111. Google Scholar [5] R. T. Glassey, J. K. Hunter and Y. Zheng, Singularities and oscillations in a nonlinear wave variational wave equation, Singularities and oscillations, The IMA Volumes in Mathematics and its applications, 91 (1997), 37-60. doi: 10.1007/978-1-4612-1972-9_3. Google Scholar [6] T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal., 63 (1977), 273-294. doi: 10.1007/BF00251584. Google Scholar [7] J. L. Johnson, Global continuous solutions of hyperbolic systems of quasilinear equations, Bull. Amer. Math. Soc., 73 (1967), 639-641. doi: 10.1090/S0002-9904-1967-11805-6. Google Scholar [8] K. Kato and Y. Sugiyama, Blow up of solutions to second sound equation in one space dimension, Kyushu J. Math., 67 (2013), 129-142. doi: 10.2206/kyushujm.67.129. Google Scholar [9] S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math., 33 (1980), 241-263. doi: 10.1002/cpa.3160330304. Google Scholar [10] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, Vol. 6, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959. Google Scholar [11] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. doi: 10.1063/1.1704154. Google Scholar [12] H. Lindblad, Global solutions of quasilinear wave equations, Amer. J. Math., 130 (2008), 115-157. doi: 10.1353/ajm.2008.0009. Google Scholar [13] R. C. MacCamy and V. J. Mizel, Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Ration. Mech. Anal., 25 (1967), 299-320. doi: 10.1007/BF00250932. Google Scholar [14] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, Appl. Math. Sci., 1984. doi: 10.1007/978-1-4612-1116-7. Google Scholar [15] R. Manfrin, A note on the formation of singularities for quasi-linear hyperbolic systems, SIAM J. Math. Anal., 32 (2000), 261-290. Google Scholar [16] M.-Y. Qi, On the Cauchy problem for a class of hyperbolic equations with initial data on the parabolic degenerating line, Acta Math. Sinica, 8 (1958), 521-529. Google Scholar [17] Y. Sugiyama, Global existence of solutions to some quasilinear wave equation in one space dimension, Differential Integral Equations, 26 (2013), 487-504. Google Scholar [18] Y. Sugiyama, Degeneracy in finite time of 1D quasilinear wave equations, SIAM J. Math. Anal., 48 (2016), 847-860. doi: 10.1137/15M1016369. Google Scholar [19] Y. Sugiyama, Large time behavior of solutions to 1D quasilinear wave equations, RIMS Kôkyûroku Bessatsu, B63 (2017), 113-123.Google Scholar [20] K. Taniguchi and Y. Tozaki, A hyperbolic equation with double characteristics which has a solution with branching singularities, Math. Japonica, 25 (1980), 279-300. Google Scholar [21] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser Boston, 1991. doi: 10.1007/978-1-4612-0431-2. Google Scholar [22] M. K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics, Micro-Local Approach, Akademie Verlag, 1997. Google Scholar [23] M. Yamaguchi and T. Nishida, On some global solution for quasilinear hyperbolic equations, Funkcial. Ekvac., 11 (1968), 51-57. Google Scholar [24] N. J. Zabusky, Exact solution for the vibrations of a nonlinear continuous model string, J. Math. Phys., 3 (1962), 1028-1039. doi: 10.1063/1.1724290. Google Scholar [25] P. Zhang and Y. Zheng, Rarefactive solutions to a nonlinear variational wave equation of liquid crystals, Comm. Partial Differential Equations, 26 (2001), 381-419. doi: 10.1081/PDE-100002240. Google Scholar [26] P. Zhang and Y. Zheng, Singular and rarefactive solutions to a nonlinear variational wave equation, Chinese Ann. Math. Series B, 22 (2001), 159-170. doi: 10.1142/S0252959901000152. Google Scholar [27] P. Zhang and Y. Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), 303-319. doi: 10.1007/s00205-002-0232-7. Google Scholar
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