International Journal of Differential Equations

Volume 2014 (2014), Article ID 724837, 5 pages

http://dx.doi.org/10.1155/2014/724837

Research Article

Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

Ülkü Dinlemez¹ and Esra Aktaş²

¹Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, Turkey

²Incirli Mahallesi, Karaelmas Sokak, Yunusemre Caddesi 51/18, İncirli, Ankara, Turkey

Received 24 December 2013; Revised 7 March 2014; Accepted 20 March 2014; Published 13 April 2014

Academic Editor: D. D. Ganji

Copyright © 2014 Ülkü Dinlemez and Esra Aktaş. This 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.

Abstract.

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 finite time.

1. Introduction

We study the damped nonlinear string equation with source term :

where , is a smooth function for with the initial conditions

and boundary conditions

The problem (1)–(3) can be regarded as modelling a nonlinear string with vertical displacement function in . And this problem has nonlinear mechanical damping of the form . The right end of the string makes it steady. The input function and the output function are applied on the left.

Wu and Li [1] studied the motion for a nonlinear beam model with nonlinear damping and external forcing terms. They 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 and source term of type . They proved global existence theorem with large initial data for . 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 . 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 first find energy equation for the problem (1)–(3). Then 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).

Lemma 1. Let and be a solution of the problem (1)–(3). Then the energy equation of the problem (1)–(3) is

Proof. Multiplying (1) with and integrating over , then we get

Applying integration by parts in the right hand side of (6), we find

And using boundary conditions in equality (7), we obtain

Hence the proof is completed.

Next we give the following theorem for global solutions in time.

Theorem 2. Assume that is a solution of the problem (1)–(3) with and

(i) satisfies the following condition:

(ii)the input and the output functions satisfy

Then the solution is global in time.

Proof. Let

Differentiating with respect to and using (5), we get

Using the Cauchy-Schwarz inequality in the last term of (12), we obtain

and it follows from (12), (13), and (10) that we have

By assumption (9) and integrating over and , respectively, we yield

Furthermore, we have

and then

Combining (11), (14), (15), and (17), we get

where . Using Gronwall’s inequality, we have

Therefore together with the continuation principle and the definition of we complete the proof of Theorem 2.

Then we give the following theorem for the blow-up solutions of the problem (1)–(3).

Theorem 3. Let be a solution of the problem (1)–(3) with . Assume that

(i)there exists such that the function satisfies

(ii)the initial values satisfy

(iii)the input and output functions satisfy

(iv) satisfies .

Then the solution blows up in finite time , and

where is some positive constant independent of the initial value and are given by (25).

Proof. We define

By virtue of (5), (21), (22), and (24), we get

Taking a derivative of (25) and using (26), we have

Multiplying (1) by and integrating over the interval and then using boundary conditions (3), we obtain

From the definition of we yield

Combining (29) and (30) in (28), we get

Using (22) in (31), we obtain

Thanks to Young's inequality,

for with and , and then we get

From embedding for and using (iv), we have and putting (34) in (32) we have

From (20), we get

Choosing and , we obtain

Thanks to (21) and (27), we yield

Now we estimate . From Holder’s inequality,

then using Young's inequality again we get

where and with . And so we have

Choosing , , we obtain

Therefore we yield

where depends on and . From (37) and (43), we have

where . Integrating (44) over , then we get

Hence blows up in finite time . is given by the inequality as below:

Consequently the solution blows up in finite time. And the proof of Theorem 3 is now finished.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and style of the paper.

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