| Title:
|
Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order (English) |
| Author:
|
Chen, Guowang |
| Author:
|
Wang, Shubin |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
36 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
475-487 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $$ u_{tt}-\left[ a_0+n a_1(u_x)^{n-1}\right]u_{xx}-a_2 u_{xxtt}=0. $$ (English) |
| Keyword:
|
nonlinear hyperbolic equation |
| Keyword:
|
initial boundary value problem |
| Keyword:
|
classical \linebreak global solution |
| Keyword:
|
blow up of solutions |
| MSC:
|
35L35 |
| MSC:
|
35L75 |
| MSC:
|
35Q72 |
| MSC:
|
74H45 |
| MSC:
|
74K10 |
| idZBL:
|
Zbl 0839.35085 |
| idMR:
|
MR1364488 |
| . |
| Date available:
|
2009-01-08T18:19:29Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118776 |
| . |
| Reference:
|
[1] Zhuang Wei, Yang Guitong: Propagation of solitary waves in the nonlinear rods.Applied Mathematics and Mechanics 7 (1986), 571-581. |
| Reference:
|
[2] Zhang Shangyuan, Zhuang Wei: Strain solitary waves in the nonlinear elastic rods (in Chinese).Acta Mechanica Sinica 20 (1988), 58-66. |
| Reference:
|
[3] 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. Zbl 0822.35094, MR 1259814 |
| Reference:
|
[4] Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.SIAM J. Math. Anal. 5 (1974), 138-146. Zbl 0243.35069, MR 0399682 |
| Reference:
|
[5] Levine H.A.: Instability & nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+F(u)$.Trans. of AMS 192 (1974), 1-21. MR 0344697 |
| . |
