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

Qingyong Gao; Fushan Li; Yanguo Wang

Open Mathematics (2011)

  • Volume: 9, Issue: 3, page 686-698
  • ISSN: 2391-5455

Abstract

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In this paper, we consider the nonlinear Kirchhoff-type equation u t t + M ( D m u ( t ) 2 2 ) ( - Δ ) m u + u t q - 2 u t = u t 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.

How to cite

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Qingyong Gao, Fushan Li, and Yanguo Wang. "Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation." Open Mathematics 9.3 (2011): 686-698. <http://eudml.org/doc/269597>.

@article{QingyongGao2011,
abstract = {In this paper, we consider the nonlinear Kirchhoff-type equation \[ u\_\{tt\} + M(\left\Vert \{D^m u(t)\} \right\Vert \_2^2 )( - \Delta )^m u + \left| \{u\_t \} \right|^\{q - 2\} u\_t = \left| \{u\_t \} \right|^\{p - 2\} u \] with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.},
author = {Qingyong Gao, Fushan Li, Yanguo Wang},
journal = {Open Mathematics},
keywords = {Blow-up; Nonlinear Kirchhoff-type equation; Positive upper bounded initial energy; nonlinear Kirchhoff-type equation; positive upper bounded initial energy; homogeneous boundary conditions},
language = {eng},
number = {3},
pages = {686-698},
title = {Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation},
url = {http://eudml.org/doc/269597},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Qingyong Gao
AU - Fushan Li
AU - Yanguo Wang
TI - Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation
JO - Open Mathematics
PY - 2011
VL - 9
IS - 3
SP - 686
EP - 698
AB - In this paper, we consider the nonlinear Kirchhoff-type equation \[ u_{tt} + M(\left\Vert {D^m u(t)} \right\Vert _2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u \] with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.
LA - eng
KW - Blow-up; Nonlinear Kirchhoff-type equation; Positive upper bounded initial energy; nonlinear Kirchhoff-type equation; positive upper bounded initial energy; homogeneous boundary conditions
UR - http://eudml.org/doc/269597
ER -

References

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  8. [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 Zbl1132.35420
  9. [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 
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  11. [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 Zbl1151.35052

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