Article
KYUNGPOOK Math. J. 2019; 59(4): 631-649
Published online December 23, 2019
Copyright © Kyungpook Mathematical Journal.
Global Nonexistence of Solutions for a Quasilinear Wave Equation with Time Delay and Acoustic Boundary Conditions
Yong Han Kang∗, Jong-Yeoul Park
Francisco College, Daegu Catholic University, Gyeongsan-si 712-702, Republic of Korea
e-mail : yonghann@cu.ac.kr
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
e-mail : jyepark@pusan.ac.kr
Received: October 8, 2019; Revised: December 4, 2019; Accepted: December 5, 2019
Abstract
In this paper, we prove the global nonexistence of solutions for a quasilinear wave equation with time delay and acoustic boundary conditions. Further, we establish the blow up result under suitable conditions.
Keywords: global nonexistence of solutions, quasilinear wave equation, blow up, time delay, acoustic boundary.
1. Introduction
In this paper, we consider the following quasilinear wave equation with time delay and acoustic boundary conditions:
Here,
The acoustic boundary conditions were introduced by Morse and Ingard [16] and developed by Beale and Rosencrans in [1], where the authors proved the global existence and regularity of the linear problem. Other authors have studied the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions (see [3, 4, 6, 7, 12, 13, 15, 19, 20, 23] and the references therein).
The time delay arises in many physical, chemical, biological and economical phenomena because these phenomena depend not only on the present state but also on the past history of the system in a more complicated way. In particular, the effects of time delay strikes on our system have a significant effect on the range of existence and the stability of the system. The differential equations with time delay effects have become an active area of research, see for example [9, 11, 17, 18]. In [14], without the delay term and the acoustic boundary condition, Liu and Wang considered the global nonexistence of solutions with the positive initial energy for a class of wave equations:
where
where
where
Motivated by the previous works, we consider an equation in a broader and more generalized form than the system discussed above. So we study the global nonexistence of solutions for a quasilinear wave equation with the time delay and acoustic boundary conditions. To the best of our knowledge. there are no results of a quasilinear wave equations with the time delay and acoustic boundary conditions. Thus the result in this work is very meaningful. The main result will be proved in Section 3.
2. Preliminaries
In this section, we shall give some notations, assumptions and a theorem which will be used throughout this paper. We denote by
We make the following same assumptions on
There are constants
for all
for all
for all values of the arguments
Remark 2.1
We note that when
Now, we transform the
Thus, we have
Then problem (
We introduce the following space
for some
We state, without a proof, a local existence which can be established by combining arguments of [2, 5, 24].
Theorem 2.1
In order to state and prove our result, we introduce the energy functional
where
We set
where
We also set
3. Proof of Main Result
In this section, we state and prove our main result. Our main result as follows.
Theorem 3.1
In this section, we shall prove Theorem 3.1. We start with a series of lemmas. We denote
Theorem 3.1 will be proved by contradiction, so we shall suppose that the solution of (
We use the idea of Vitillaro [22].
Lemma 3.1
Multiplying the
On the other hand, we have from the equation in (
Also, multiplying the
and
Hence, from (
By using (
Hence we get
Lemma 3.2
First, we will prove the (i). From (
Using (
and then
Therefore
where
This is impossible since
Thus, the proof is complete.
In the remainder of this section, we consider initial values (
Then we have the following Lemma.
Lemma 3.3
From Lemma 3.1, we see that
From (
From (
Thus, combing (
Now, we define
for
By taking a derivative of (
By using (
Exploiting Hölder’s and Young’s inequality and
By Young’s inequality, we get
A substitution of (
Therefore, we choose
for
If
From (
we have (see[15])
and
for all
for some constant
From (
From (
and
Thus, it follows that
At this point, choosing
we deduce
where
Then from(
and
On the other hand, from(
Then the above inequality leads to
Next, using Hölder’s inequality, the embedding
From (
Therefore, there exists a positive constant
Furthermore, by the same method, we deduce
Similarly, we find
Using the embedding
where
Using Young’s inequality, we write
where
where
where
Integration of (
Therefore
Thus the proof of Theorem 2.1 is complete.
Acknowledgements
We would like to thank the references for their valuable comments and suggestions to improve our paper.
Funding
Y. H. Kang was supported by research grants from the Daegu Catholic University in 2017(Number 20171288).
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Competing interests
The authors declare that they have no competing interests.
Abbreviations
Not applicable.
Author’s contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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