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Blow-up of solutions to a nonlinear dispersive rod equation. (English) Zbl 1172.35504
Summary: In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions.
MSC:
35Q72Other PDE from mechanics (MSC2000)
35B40Asymptotic behavior of solutions of PDE
35G25Initial value problems for nonlinear higher-order PDE
74B20Nonlinear elasticity
74H20Existence of solutions for dynamical problems in solid mechanics
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
References:
[1]Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272(1220), 47–78 (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
[2]Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons, Phys. Rev. Letter. 71, 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[3]Constantin, A.: On the Cauchy problem for the periodic Camassa-Holm equation. J. Differential Equations 141(2), 218–235 (1997) · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[4]Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50(2), 321–362 (2000)
[5]Constantin, A., Escher, J.: Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51, 475–504 (1998) · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[6]Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181, 229–243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[7]Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. 211(1), 45–61 (2000) · Zbl 1002.35101 · doi:10.1007/s002200050801
[8]Constantin, A., McKean, H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52(8), 949–982 (1999) · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[9]Constantin, A., Strauss, W.: Stability of peakons, Comm. Pure Appl. Math. 53, 603–610 (2000) · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[10]Constantin, A., Strauss, W.: Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A 270(3–4), 140–148 (2000) · Zbl 1115.74339 · doi:10.1016/S0375-9601(00)00255-3
[11]Dai, H.-H.: Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech. 127(1–4), 193–207 (1998)
[12]Dai, H.-H., Huo, Y.: Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(1994), 331–363 (2000) · Zbl 1004.74046 · doi:10.1098/rspa.2000.0520
[13]Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Backlund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981/82) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[14]Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987) · Zbl 0656.35122 · doi:10.1016/0022-1236(87)90044-9
[15]Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[16]Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jorgens), pp. 25–70. Lecture Notes in Math., Vol. 448, Springer, Berlin (1975)
[17]Li, Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations 162, 27–63 (2000)
[18]McKean, H.P.: Breakdown of a shallow water equation, Asian J. Math. 2(4), 867–874 (1998)
[19]Misioł ek, G.: Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal. 12(5), 1080–1104 (2002) · Zbl 1158.37311 · doi:10.1007/PL00012648
[20]Molinet, L.: On well-posedness results for Camassa-Holm equation on the line: a survey. J. Nonlinear Math. Phys. 11(4), 521–533 (2004) · Zbl 1069.35076 · doi:10.2991/jnmp.2004.11.4.8
[21]Seliger, R.: A note on the breaking of waves. Proc. Roy. Soc. Lond Ser. A. 303, 493–496 (1968) · Zbl 0159.28502 · doi:10.1098/rspa.1968.0063
[22]Shkoller, S.: Geometry and curvature of diffeomorphism groups with H1 metric and mean hydrodynamics, J. Funct. Anal. 160(1), 337–365 (1998) · Zbl 0933.58010 · doi:10.1006/jfan.1998.3335
[23]Struwe, M.: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Second edition. Results in Mathematics and Related Areas (3), 34. Springer-Verlag, Berlin (1996)
[24]Xin, Z., Zhang, P.: On the weak solution to a shallow water equation, Comm. Pure Appl. Math. 53, 1411–1433 (2000) · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[25]Xin, Z., Zhang, P.: On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Comm. Partial Differential Equations 27(9–10), 1815–1844 (2002) · Zbl 1034.35115 · doi:10.1081/PDE-120016129
[26]Zhou, Y.: Wave breaking for a periodic shallow water equation. J. Math. Anal. Appl. 290, 591–604 (2004) · Zbl 1042.35060 · doi:10.1016/j.jmaa.2003.10.017
[27]Zhou, Y.: Stability of solitary waves for a rod equation. Chaos Solitons & Fractals 21(4), 977–981 (2004) · Zbl 1046.35094 · doi:10.1016/j.chaos.2003.12.030
[28]Zhou, Y.: Well-posedness and blow-up criteria of solutions for a rod equation. Math. Nachr. 278, 1726–1739 (2005) · Zbl 1125.35103 · doi:10.1002/mana.200310337
[29]Zhou, Y.: Blow-up phenomenon for a periodic rod equation. Submitted (2004)