# Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order

Commentationes Mathematicae Universitatis Carolinae (1995)

- Volume: 36, Issue: 3, page 475-487
- ISSN: 0010-2628

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topChen, Guo Wang, and Wang, Shu Bin. "Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 475-487. <http://eudml.org/doc/247746>.

@article{Chen1995,

abstract = {The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation \[ u\_\{tt\}-\alpha u\_\{xx\}-\beta u\_\{xxtt\}=\varphi (u\_x)\_x \]
are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods \[ u\_\{tt\}-\left[ a\_0+n a\_1(u\_x)^\{n-1\}\right]u\_\{xx\}-a\_2 u\_\{xxtt\}=0. \]},

author = {Chen, Guo Wang, Wang, Shu Bin},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {nonlinear hyperbolic equation; initial boundary value problem; classical global solution; blow up of solutions; global solution; blow up},

language = {eng},

number = {3},

pages = {475-487},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order},

url = {http://eudml.org/doc/247746},

volume = {36},

year = {1995},

}

TY - JOUR

AU - Chen, Guo Wang

AU - Wang, Shu Bin

TI - Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1995

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 36

IS - 3

SP - 475

EP - 487

AB - The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation \[ u_{tt}-\alpha u_{xx}-\beta u_{xxtt}=\varphi (u_x)_x \]
are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods \[ u_{tt}-\left[ a_0+n a_1(u_x)^{n-1}\right]u_{xx}-a_2 u_{xxtt}=0. \]

LA - eng

KW - nonlinear hyperbolic equation; initial boundary value problem; classical global solution; blow up of solutions; global solution; blow up

UR - http://eudml.org/doc/247746

ER -

## References

top- Zhuang Wei, Yang Guitong, Propagation of solitary waves in the nonlinear rods, Applied Mathematics and Mechanics 7 (1986), 571-581. (1986)
- Zhang Shangyuan, Zhuang Wei, Strain solitary waves in the nonlinear elastic rods (in Chinese), Acta Mechanica Sinica 20 (1988), 58-66. (1988)
- Chen Guowang, Yang Zhijian, Zhao Zhancai, Initial value problems and first boundary problems for a class of quasilinear wave equations, Acta Mathematicae Applicate Sinica 9 (1993), 289-301. (1993) Zbl0822.35094MR1259814
- Levine H.A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138-146. (1974) Zbl0243.35069MR0399682
- Levine H.A., Instability nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=-Au+F\left(u\right)$, Trans. of AMS 192 (1974), 1-21. (1974) MR0344697

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