Geodesic
Blowup of a positive-energy solution of model wave equations in nonlinear dynamics
Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 1, pp. 3-17 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider four problems of model nonlinear equations appearing in nonlinear mechanics and obtain sufficient conditions for the finite-time blowup of the problem solutions in bounded domains with homogeneous Dirichlet conditions. The initial system energy can be an arbitrarily large positive quantity. We use a modified Levin method to prove the blowup.
Keywords: finite-time blowup, generalized Klein–Gordon equation, nonlinear hyperbolic equation, nonlinear mixed boundary value problem, field theory. 
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M. O. Korpusov. Blowup of a positive-energy solution of model wave equations in nonlinear dynamics. Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 1, pp. 3-17. http://geodesic.mathdoc.fr/item/TMF_2012_171_1_a0/

[1] K. Li, Q. Zhang, J. Differ. Equations, 146:1 (1998), 5–21 | DOI | MR | Zbl

[2] V. I. Erofeev, V. V. Kazhaev, N. P. Semerikova, Volny v sterzhnyakh. Dispersiya. Dissipatsiya. Nelineinost, Fizmatlit, M., 2002

[3] G. Kauderer, Nelineinaya mekhanika, IL, M., 1961 | MR

[4] Y. Wang, IMA J. Appl. Math., 74:3 (2009), 392–415 | DOI | MR | Zbl

[5] A. B. Al'shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Analysis and Applications, 15, Walter de Gruyter, Berlin, 2011 | MR | Zbl

[6] H. A. Levine, Trans. Amer. Math. Soc., 192 (1974), 1–21 | MR | Zbl

[7] H. A. Levine, P. Pucci, J. Serrin, Contemp. Math., 208 (1997), 253–263 | DOI | MR | Zbl

[8] P. Pucci, J. Serrin, Topol. Methods Nonlinear Anal., 10:2 (1997), 241–247 | DOI | MR | Zbl

[9] P. Pucci, J. Serrin, J. Differ. Equations, 150:1 (1998), 203–214 | DOI | MR | Zbl

[10] B. Straughan, Proc. Amer. Math. Soc., 48 (1975), 381–390 | DOI | MR | Zbl

[11] H. A. Levine, G. Todorova, Proc. Amer. Math. Soc., 129 (2003), 793–805 | DOI | MR

[12] E. L. Mitidieri, S. I. Pokhozhaev, Tr. MIAN, 234 (2001), 3–383 | MR | Zbl

[13] S. I. Pohozaev, Milan J. Math., 77:1 (2009), 127–150 | DOI | MR | Zbl

[14] V. A. Galaktionov, S. I. Pohozaev, Blow-up, critical exponents and asymptotic spectra for nonlinear hyperbolic equations, Preprint 00/10, Univ. of Bath, Bath, 2000 | MR

[15] V. A. Galaktionov, S. I. Pokhozhaev, Zh. vychisl. matem. i matem. fiz., 48:10 (2008), 1819–1846 | DOI | MR | Zbl

[16] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[17] V. K. Kalantarov, O. A. Ladyzhenskaya, Zap. nauchn. sem. LOMI, 69 (1977), 77–102 | DOI | MR | Zbl

[18] T. Cazenave, A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, Oxford Univ. Press, Oxford, 1998 | MR | Zbl

[19] V. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR | Zbl

[20] O. A. Lima, A. T. Lourêndo, A. O. Marinho, Electron. J. Differ. Equations, 130 (2006), 1–18 | MR | Zbl

[21] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR | Zbl

[22] Yu. A. Dubinskii, Matem. sb., 67(109):4 (1965), 609–642 | MR