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2011 | 9 | 3 | 686-698
Tytuł artykułu

Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation

Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
In this paper, we consider the nonlinear Kirchhoff-type equation $$ u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u $$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
3
Strony
686-698
Opis fizyczny
Daty
wydano
2011-06-01
online
2011-03-22
Twórcy
autor
autor
autor
  • Qufu Normal University
Bibliografia
  • [1] Adams R.A., Sobolev Spaces, Pure Appl. Math., 65, Academic Press, New York-London, 1975
  • [2] Chen W., Zhou Y., Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal., 2009, 70(9), 3203–3208 http://dx.doi.org/10.1016/j.na.2008.04.024
  • [3] Georgiev V., Todorova G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 1994, 109(2), 295–308 http://dx.doi.org/10.1006/jdeq.1994.1051
  • [4] Kirchhoff G., Vorlesungen über Mechanik, 3rd ed., Teubner, Leipzig, 1883
  • [5] Levine H.A., Park S.R., Serrin J., Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 1998, 228(1), 181–205 http://dx.doi.org/10.1006/jmaa.1998.6126
  • [6] Li F.C., Global existence and blow-up of solutions for a higher-order Kirchhoff-type equation with nonlinear dissipation, Appl. Math. Lett., 2004, 17(12), 1409–1414 http://dx.doi.org/10.1016/j.am1.2003.07.014
  • [7] Messaoudi S.A., Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 2002, 265(2), 296–308 http://dx.doi.org/10.1006/jmaa.2001.7697
  • [8] Messaoudi S.A., Said Houari B., A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation, Appl. Math. Lett., 2007, 20(8), 866–871 http://dx.doi.org/10.1016/j.aml.2006.08.018
  • [9] Ono K., On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 1997, 216(1), 321–342 http://dx.doi.org/10.1006/jmaa.1997.5697
  • [10] Vitillaro E., Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 1999, 149(2), 155–182 http://dx.doi.org/10.1007/s002050050171
  • [11] Wu S.T., Tsai L.Y., Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation, Nonlinear Anal., 2006, 65(2), 243–264 http://dx.doi.org/10.1016/j.na.2004.11.023
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-010-0096-2
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