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Abstrakty
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.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
686-698
Opis fizyczny
Daty
wydano
2011-06-01
online
2011-03-22
Twórcy
autor
- Qufu Normal University, gaoqingyong1219@163.com
autor
- Qufu Normal University, fushan99@163.com
autor
- Qufu Normal University
Bibliografia
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- [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
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-010-0096-2