Abstract
We consider the Dirichlet problem for a nonlinear system of equations, continuing our study of nonlinear hyperbolic equations and systems of equations with an arbitrarily large positive energy. We use a modified Levine method to prove the blow-up.
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References
I. Segal, Proc. Symp. Appl. Math. Amer. Math. Soc., 17, 210–226 (1965).
J. Zhang, Math. Methods Appl. Sci., 26, 11–25 (2003).
Y. Wang, IMA J. Appl. Math., 74, 392–415 (2009).
W. Liu, Nonlinear Anal., 73, 244–255 (2010).
A. B. Al’shin, M. O. Korpusov, and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations (De Gruyter Ser. Nonlinear Anal. Appl., Vol. 15), De Gruyter, Berlin (2011).
H. A. Levine, Trans. Amer. Math. Soc., 192, 1–21 (1974).
H. A. Levine, P. Pucci, and J. Serrin, “Some remarks on global nonexistence for nonautonomous abstract evolution equations,” in: Harmonic Analysis and Nonlinear Differential Equations: A Volume in Honor of Victor L. Shapiro (Contemp. Math., Vol. 208, M. L. Lapidus, L. H. Harper, and A. J. Rumbos, eds.), Amer. Math. Soc., Providence, R. I. (1997), pp. 253–263.
P. Pucci and J. Serrin, Topol. Methods Nonlinear Anal., 10, 241–247 (1997).
P. Pucci and J. Serrin, J. Differential Equations, 150, 203–214 (1998).
B. Straughan, Proc. Amer. Math. Soc., 48, 381–390 (1975).
V. K. Kalantarov and O. A. Ladyzhenskaya, J. Soviet Math., 10, 53–70 (1978).
H. A. Levine and G. Todorova, Proc. Amer. Math. Soc., 129, 793–805 (2003).
E. Mitidieri and S. I. Pokhozhaev, A Priori Estimates and Blow-Up of Solutions of Nonlinear Partial Differential Equations and Inequalities [in Russian] (Trudy Mat. Inst. Steklova, Vol. 234), Nauka, Moscow (2001).
S. I. Pohozaev, Milan J. Math., 77, 127–150 (2009).
V. A. Galaktionov and S. I. Pohozaev, “Blow-up, critical exponents, and asymptotic spectra for nonlinear hyperbolic equations,” Preprint 00/10, Univ. of Bath, Bath, UK (2000).
V. A. Galaktionov and S. I. Pokhozhaev, Comput. Math. Math. Phys., 48, 1784–1810 (2008).
A. A. Samarskij, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhajlov, Peaking Modes in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1987).
M. O. Korpusov and A. G. Sveshnikov, Sb. Math., 200, 549–572 (2009).
J.-L. Lions, Quelques méthodes de résolution des probl`emes aux limites non linéaires, Gauthier-Villars, Paris (1969).
B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).
O. A. Lima, A. T. Lourdo, and A. O. Marinho, Electronic J. Differential Equations, 130, 1–18 (2006).
Yu. A. Dubinskii, Mat. Sb., n.s., 67(109), 609–642 (1965).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 171, No. 3, pp. 355–369, June, 2012.
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Korpusov, M.O. Blow-up of the solution of a nonlinear system of equations with positive energy. Theor Math Phys 171, 725–738 (2012). https://doi.org/10.1007/s11232-012-0070-1
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DOI: https://doi.org/10.1007/s11232-012-0070-1