KYUNGPOOK Math. J. 2019; 59(4): 631-649  
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 :
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
e-mail :
* Corresponding Author.
Received: October 8, 2019; Revised: December 4, 2019; Accepted: December 5, 2019; Published online: December 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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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:

(ut(x,t)l-2ut(x,t))t-Δut(x,t)-div(a(x)u(x,t)α-2u(x,t))-div(ut(x,t)β-2ut(x,t))+Q(x,t,ut)+μ1ut(x,t)+μ2ut(x,t-τ)=f(x,u(x,t))in Ω×[0,T),u=0on Γ0×[0,T),ut(x,t)ν+a(x)u(x,t)α-2ut(x,t)ν+ut(x,t)β-2ut(x,t)ν=h(x)yt(x,t)on Γ1×[0,T),ut(x,t)+k(x)yt(x,t)+q(x)y(x,t)=0on Γ1×[0,T),u(x,0)=u0(x),ut(x,0)=u1(x)in Ω,ut(x,t-τ)=f0(x,t-τ)in Ω×(0,τ),y(x,0)=y0(x)on Γ1.

Here, J = [0, T), 0 < T ≤ ∞, a: Ω → R+ is a positive function, l, α, β ≥ 2, μ1 > 0, μ2 is a real number, and τ > 0 represents the time delay. Further, Ω is a regular and bounded domain of Rn(n ≥ 1) and ∂Ω(:= Γ) = Γ0 ∪ Γ1, where Γ0 and Γ1 are closed and disjoint and ν denotes the outer normal derivative. The functions k, q, h: Γ1R+(:= [0,∞]) are essentially bounded and 0 < q0q(x) on Γ1.

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:

(ut(x,t)l-2ut(x,t))t-Δut(x,t)-div(a(x)u(x,t)α-2u(x,t))-div(u(x,t)β-2ut(x,t))+Q(x,t,ut)=f(x,u(x,t))in J×Ω,u(x,t)=0on J×Ω,u(x,0)=u0(x),ut(x,0)=u1(x)in Ω,

where J = [0, T), 0 < T ≤ ∞, Ω is a bounded regular open subset of Rn(n ≥ 1), l, α, β ≥ 2 and a, Q, f satisfy some conditions. Recently, for l = 2, a(x) = 1, Q(ut) = a|ut|m−2ut, μ1 = μ2 = 0, f(u) = b|u|p−2u, and without the time delay term in our system, Jeong at al [8] investigated the global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions

utt-Δut-div(uα-2u)-div(utβ-2ut)+αutm-2ut=bup-2uin Ω×(0,),u=0on Γ0×(0,),utν+uα-2uν+utβ-2utν=h(x)yton Γ1×(0,),ut+f(x)yt+q(x)y=0on Γ1×(0,),u(x,0)=u0(u),ut(x,0)=u1(x)in Ω,y(x,0)=y0(x)on Γ1,

where a, b > 0, α, β, m, p > 2 are constants and Ω is a regular and bounded domain of Rn(n ≥ 1) and ∂Ω(= Γ) = Γ0 ∪ Γ1. Here Γ0 and Γ1 are closed and disjoint. The functions h, f, q: Γ1R+ are essentially bounded. Moreover, for a(x) = 1, l = 2, div(|∇ut|β−2ut) = 0, Q = 0, and without boundary conditions, Kafini and Messaoudi [10] studied the following nonlinear damped wave equation

utt(x,t)-div(u(x,t)m-2u(x,t))+μ1ut(x,t)+μ2ut(x,t-τ)=bu(x,t)p-2u(x,t)in Ω×(0,),ut(x,t-τ)=f0(x,t-τ)on (0,τ),u(x,t)=0on Ω×(0,),u(x,0)=u0(x),ut(x,0)=u1(x)in Ω,

where p > m ≥ 2, b, μ1 are positive constants, μ2 is a real number, and τ > 0 represents the time delay. They proved the blow-up result in a nonlinear wave equation with time delay and without acoustic boundary conditions.

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 m′ the Hölder conjugate of m, ||u||p = ||u||Lp(Ω), ||u||p = ||u||Lp(Γ), ||u||1,s = ||u||W1,s(Ω), where Lp(Ω) and W1,s(Ω) stand for the Lebesgue spaces and the classical Sobolev spaces, respectively. Specially we introduce the set

WΓ01,s(Ω)={uW1,su=0on Γ0},W01,s(Ω)={uW1,su=0on Γ}.

We make the following same assumptions on a, Q, f as section 4.2 of [22].

(H1)a(x) ∈ L(Ω) such that a(x) ≥ a0 a.e. in Ω for some a0 > 0.

(H2)f(x, u) ∈ C(Ω×ℝn,ℝn) and f(x, u) = ∇uΦ(x, u), with normalizing condition Φ(x, 0) = 0.

There are constants d1 > 0, p > α and μ < μ0a0 such that


for all x ∈ Ω and u ∈ ℝn. Moreover, there is ε1 > 0 such that for all ε ∈ (0, ε1] there exists d2 = d2(ε) > (pα)d1/p such that


for all x ∈ Ω.

(H3) There are m > 1 and a measurable function d = d(x, t) defined on Ω×J such that d(·, t) ∈ Lp/(pm)(Ω) for a.e. tJ and


for all values of the arguments x, t, v, where


Remark 2.1

We note that when Q(x, t, ut) = b(1+t)ρ|ut|m−2ut, −∞ < ρm−1, condition (H3) holds.

Now, we transform the equation (1.1)(1.7) to the system, using the idea of [21] and introduce the associated energy. So, we introduce the new variable:

z(x,ρ,t)=ut(x,t-τρ),xΩ,   ρ(0,1),   t>0.

Thus, we have

τzt(x,ρ,t)+zρ(x,ρ,t)=0,xΩ,   ρ(0,1),   t>0.

Then problem (1.1)–(1.7) takes the following form:

(ut(x,t)l-2ut(x,t))t-Δut(x,t)-div(a(x)u(x,t)α-2u(x,t))-div(ut(x,t)β-2ut(x,t))+Q(x,t,ut)+μ1ut(x,t)+μ2z(x,1,t)=f(x,u(x,t))in Ω×J,τzt(x,ρ,t)+zρ(x,ρ,t)=0         in         Ω×(0,1)×J,u=0on Γ0×J,ut(x,t)ν+a(x)u(x,t)α-2u(x,t)ν+ut(x,t)β-2ut(x,t)ν=h(x)yt(x,t)on Γ1×J,ut(x,t)+k(x)yt(x,t)+q(x)y(x,t)=0on Γ1×J,u(x,0)=u0(x),ut(x,0)=u1(x)in Ω,z(x,ρ,0)=f0(x,-ρτ)in Ω×(0,1),y(x,0)=y0(x)on Γ1.

We introduce the following space


for some T > 0.

We state, without a proof, a local existence which can be established by combining arguments of [2, 5, 24].

Theorem 2.1

Letu0WΓ01,α(Ω), u1L2(Ω), f0L2(Ω×(0, 1)) and y0L21) be given. Suppose that l, α, β, m, p > 2, max{l, β, m} < α < p < nα/(nα), μ1 > |μ2| and (H1)(H3) hold. Then problem(2.6)–(2.13) has a unique local solution (u, z, y) ∈ Z×L2([0, T);L2(Ω×(0, 1)))×L2([0, T);L21)) for some T > 0.

In order to state and prove our result, we introduce the energy functional




We set


where B1 is the best constant of the Sobolev embedding W01,α(Ω)Lp(Ω) given by


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

Letu0WΓ01,α(Ω), u1L2(Ω), f0L2(Ω×(0, 1)) and y0L21) be given. Suppose that l, α, β, m, p > 2, max{l, β, m} < α < p < nα/(nα), μ1 > |μ2| and (H1)(H3) hold. Assume further that


Then the solution (u, z, y) ∈ Z × L2(R+);L2(Ω × (0, 1))) × L2(R+);L21)) of problem (2.6)–(2.13) can not exist for all time.

In this section, we shall prove Theorem 3.1. We start with a series of lemmas. We denote

λ0=u0α,         E0=E(0).

Theorem 3.1 will be proved by contradiction, so we shall suppose that the solution of (2.6)–(2.13) exists on the whole interval [0,∞), i.e. T = ∞.

Proof of Theorem 3.1

We use the idea of Vitillaro [22].

Lemma 3.1

Let (u, z, y) be the solution of (2.6)–(2.13). Then the energy functional defined by (2.15) satisfies, for some constant c0 > 0,


Multiplying the equation (2.6) by ut(t), integrating over Ω, using Green’s formula and exploiting the equation (2.9), we obtain


On the other hand, we have from the equation in (2.10) that


Also, multiplying the equation (2.7) by ξ2z(x,ρ,t) and integrating over Ω × (0, 1), we deduce




Hence, from (2.15) and (3.3)–(3.6), we arrive at


By using (2.16), we get, for some c0 > 0,


Hence we get E(t) ≤ E(0) for all tJ.

Lemma 3.2

If (λ0, E(0)) ∈ ∑, then we have

(i)         u(t)αλ2foralltJ,forsomeλ2>λ1,(ii)         u(t)pB1λ2foralltJ,forthesomeλ2in(i).

First, we will prove the (i). From (2.15), we see that


Using (2.1), since f(x, u(t)) = ∇uΦ(x, u(t)), it follows that


and then




where λ = ||∇u(t)||α. It is easy to verify that ϕ is increasing for 0 < λ < λ1, decreasing for λ > λ1, ϕ(λ) → −∞ as λ → +∞ and ϕ(λ) = E1, where λ1 is given in (2.18). Therefore, since E0< E1, there exists λ2> λ1 such that ϕ(λ2) = E(0). From (3.12) we have ϕ(λ0) ≤ E(0) = ϕ(λ2), which implies that λ0λ2 since λ0 > λ1. To proof the result, we suppose by contradiction that ||∇u0||α< λ2, for some t0 > 0 and by the continuity of ||∇u(t)||α we can choose such that ||∇u(t0)||α> λ1. Again the use of (3.12) leads to


This is impossible since E(t) ≤ E(0), for all t ≥ 0. Thus (i) is established. Next, we will prove the (ii). From (3.12), we get


Thus, the proof is complete.

In the remainder of this section, we consider initial values (λ0, E0) ∈ ∑. We set


Then we have the following Lemma.

Lemma 3.3

For all tJ, we have


From Lemma 3.1, we see that H′(t) ≥ 0. Thus, we deduce


From (3.12), we obtain


From (3.8), ||∇u(t)||α> λ1, we get


Thus, combing (3.15) and (3.16) we obtain (3.14).

Now, we define


for ɛ small to be chosen later and


By taking a derivative of (3.17) we have


By using (2.6)–(2.10) and estimate (3.19), we find


Exploiting Hölder’s and Young’s inequality and (H3), for any δ, μ, η, ρ > 0, we deduce


By Young’s inequality, we get


A substitution of (3.21) – (3.24) into (3.20) yields


Therefore, we choose δ, μ, η, and ρ so that


for M1, M2, M3, M4 to be specified later. Using (2.10), (3.25) and (3.26), we arrive at


If M=M2+(β-1)M3β+(m-1)M1m+μ2M4, then (3.27) takes the form


From (3.14),(3.18), the embedding W1,α(Ω) ↪ Lp(Ω) and


we have (see[15])




for all t ≥ 0, where d = c(Ω)[1 + 1/H(0)]. Inserting estimates (3.29)–(3.32) into (3.28), we obtain


for some constant k and

c1=cdm(B1pd1p)σ/(m-1)B1m,         c2=d4(B1pd1p)σ,c3=dβ(B1pd1p)σ(β-1),         c4=d(B1pd1p)σB12.

From (2.17),(2.18) and Lemma 3.2, we have


From (2.2), we can choose k satisfying

αɛk<pɛmin {αd2(p-α)d1,1}



Thus, it follows that


At this point, choosing M1, M2, M3, M4 large enough and ɛ sufficiently small and using


we deduce


where γ is a positive constant (it is possible since k > ɛα). We choose ɛ sufficiently small and 0 < ɛ < (1 – σ)/M so that


Then from(3.33) we get

L(t)L(0)0,         t0,



On the other hand, from(3.17) and h(x), q(x) > 0, we have


Then the above inequality leads to


Next, using Hölder’s inequality, the embedding W1,α(Ω) ↪ Ll(Ω), α > l and Young’s inequality, we derive


From (3.18) and (3.29), we obtain


Therefore, there exists a positive constant C′ such that for all t ≥ 0,


Furthermore, by the same method, we deduce


Similarly, we find


Using the embedding W01,α(Ω)L2(Γ1) and Hölder’s inequality, we get


where c5 is a embedding constant. Consequently, there exists a positive constant c6 = c(||h||, ||q||, q0, σ, α) such that


Using Young’s inequality, we write


where c7 is a positive constant depending on c6 and α. Applying once again the algebraic inequality (3.29) with z=u(t)αα, ν = 2/[α(1 – 2σ)] and making use of (3.18), we see that by the same method as above


where c8 is a positive constant. Hence combining (3.35) – (3.37) and using α > 2, we arrive at


where C* is a positive constant. Consequently a combining of (3.34) and (3.38), for some ξ > 0, we obtain


Integration of (3.9) over (0, t) yield


Therefore L(t) blow up in finite time


Thus the proof of Theorem 2.1 is complete.


We would like to thank the references for their valuable comments and suggestions to improve our paper.


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.


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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