Research Article
Global and Blow-Up Solutions for Nonlinear Hyperbolic
Equations with Initial-Boundary Conditions
Ülkü Dinlemez
1
and Esra AktaG
2
1
Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, Turkey
2
Incirli Mahallesi, Karaelmas Sokak, Yunusemre Caddesi 51/18,
˙
Incirli, Ankara, Turkey
Correspondence should be addressed to
¨
Ulk¨ u Dinlemez; ulku@gazi.edu.tr
Received 24 December 2013; Revised 7 March 2014; Accepted 20 March 2014; Published 13 April 2014
Academic Editor: D. D. Ganji
Copyright © 2014
¨
U. Dinlemez and E. Aktas ¸. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under
suitable conditions the solution is global in time and the solution with a negative initial energy blows up in fnite time.
1. Introduction
We study the damped nonlinear string equation with source
term ||
:
+
=((
2
)
)
+ ||
,
(, ) ∈ (0, 1) × [0, ] ,
(1)
where 1<, () is a smooth function for 0≤ with the
initial conditions
(,0)=
0
(),
(, 0) =
1
(), ∈ [0, 1] , (2)
and boundary conditions
(1, ) = 0, ∈(0,),
(
(0, )
2
(0, ))−
(0, ) = 2 () , ∈[0,],
(
(0, )
2
(0, ))+
(0, ) = 2 () , ∈ [0, ] .
(3)
Te problem (1)–(3) can be regarded as modelling a nonlinear
string with vertical displacement function (, ) in R. And
this problem has nonlinear mechanical damping of the form
||
. Te right end of the string makes it steady. Te input
() function and the output () function are applied on the
lef.
Wu and Li [1] studied the motion for a nonlinear
beam model with nonlinear damping |
|
−1
and external
forcing ||
−1
terms. Tey showed that this model has
a unique global solution and blow-up solution under the
same conditions. Levine et al. [2] and Levine and Serrin [3]
studied abstract version. Georgiev and Todorova [4] studied
nonlinear wave equations involving the nonlinear damping
term |
|
−1
and source term of type |
|
−1
. Tey proved
global existence theorem with large initial data for 1<≤.
Hao and Li [5] studied the global solutions for a nonlinear
string with boundary input and output. Dinlemez [6] proved
the global existence and uniqueness of weak solutions for
the initial-boundary value problem for a nonlinear wave
equation with strong structural damping and nonlinear
source terms in R. A lot of papers in connection with blow-
up, global solutions and existence of weak solutions were
studied in [7–15].
In this paper we frst fnd energy equation for the problem
(1)–(3). Ten we prove the solutions of the problem (1)–(3)
are global in time under some conditions on the function
(), input (), and the output (). Finally we establish a
blow-up result for solutions with a negative initial energy. Our
approach is similar to the one in [5].
2. Main Results
Now we give the following lemma for energy equation for the
problem (1)–(3).
Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2014, Article ID 724837, 5 pages
http://dx.doi.org/10.1155/2014/724837