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Blow-up phenomenon for a periodic rod equation. (English) Zbl 1181.35287

Summary: We consider a new rod equation derived recently by H.-H. Dai [Acta Mech. 127, No. 1–4, 193–207 (1998; Zbl 0910.73036)] for a compressible hyperelastic material. We explore various sufficient conditions of the initial data which guarantee the blow-up in finite time for periodic case. The focus of this Letter is on two particular classes of initial data, \(\int_{\mathbb S}u_0=0\) or \(u_{0}\) is odd.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics

Citations:

Zbl 0910.73036
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References:

[1] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Philos. Trans. R. Soc. London Ser. A, 272, 47 (1972)
[2] Camassa, R.; Holm, D., Phys. Rev. Lett., 71, 1661 (1993)
[3] Constantin, A., Ann. Inst. Fourier (Grenoble), 50, 321 (2000)
[4] Constantin, A.; Escher, J., Commun. Pure Appl. Math., 51, 475 (1998)
[5] Constantin, A.; Escher, J., Acta Math., 181, 229 (1998)
[6] Constantin, A.; Escher, J., Indiana Univ. Math. J., 47, 1527 (1998)
[7] Constantin, A.; Molinet, L., Commun. Math. Phys., 211, 45 (2000)
[8] Constantin, A.; Strauss, W., Commun. Pure Appl. Math., 53, 603 (2000)
[9] Constantin, A.; Strauss, W., Phys. Lett. A, 270, 140 (2000)
[10] Dai, H.-H., Acta Mech., 127, 193 (1998)
[11] Dai, H.-H.; Huo, Y., R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci., 456, 331 (2000)
[12] Grillakis, M.; Shatah, J.; Strauss, W., J. Funct. Anal., 74, 160 (1987)
[13] Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations. Spectral theory and differential equations, (Proc. Sympos. Dedicated to Konrad Jorgens. Proc. Sympos. Dedicated to Konrad Jorgens, Dundee, 1974. Proc. Sympos. Dedicated to Konrad Jorgens. Proc. Sympos. Dedicated to Konrad Jorgens, Dundee, 1974, Lecture Notes in Mathematics, vol. 448 (1975), Springer: Springer Berlin), 25-70
[14] Li, Y.; Olver, P., J. Differential Equations, 162, 27 (2000)
[15] McKean, H. P., Asian J. Math., 2, 867 (1998)
[16] Misiołek, G., Geom. Funct. Anal., 12, 1080 (2002)
[17] Molinet, L., J. Nonlinear Math. Phys., 11, 521 (2004)
[18] Seliger, R., Proc. R. Soc. London Ser. A, 303, 493 (1968)
[19] Shkoller, S., J. Funct. Anal., 160, 337 (1998)
[20] Xin, Z.; Zhang, P., Commun. Pure Appl. Math., 53, 1411 (2000)
[21] Xin, Z.; Zhang, P., Commun. Partial Differential Equations, 27, 1815 (2002)
[22] Zhou, Y., J. Math. Anal. Appl., 290, 591 (2004)
[23] Zhou, Y., Math. Nachr., 278, 1726 (2005)
[24] Zhou, Y., Chaos Solitons Fractals, 21, 977 (2004)
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