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

Open Mathematics (2011)

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

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Abstract

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In this paper, we consider the nonlinear Kirchhoff-type equation ${u}_{tt}+M\left({∥{D}^{m}u\left(t\right)∥}_{2}^{2}\right){\left(-\Delta \right)}^{m}u+{\left|{u}_{t}\right|}^{q-2}{u}_{t}={\left|{u}_{t}\right|}^{p-2}u$ with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.

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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