# Existence, Decay, and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Delay Term

Authors: Hazal Yüksekkaya; Erhan Pişkin; Salah Mahmoud Boulaaras; Bahri Belkacem Cherif
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This article deals with the study of the higher-order Kirchhoff-type equation with delay term in a bounded domain with initial boundary conditions, where firstly, we prove the global existence result of the solution. Then, we discuss the decay of solutions by using Nakao’s technique and denote polynomially and exponentially. Furthermore, the blow-up result is established for negative initial energy under appropriate conditions.


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## Body

## 1. Introduction

In this paper, we establish the higher-order Kirchhoff-type equation with delay term as follows:(1)utt+∫ΩAm/2u2dxqAmu+μ1μtx,tr−1μtx,t+μ2μtx,t−τr−1utx,t−τ=up−1u,x,t∈Ω×0,T,ux,0=u0x,utx,0=u1xx∈Ωutx,t−τ=f0x,t−τx∈Ω,t∈0,τ,∂iu∂vi=0,i=0,1,⋯,m−1x∈∂Ω,whereA=−Δ,m≥1is a natural number, q,r≥0 are real numbers, p>1 is a real number and is a bounded domain with smooth boundary ∂Ω in Rn, n=1;2;3; v is the outer normal. τ>0 denotes time delay, and 1 and 2 are positive real numbers. The functions u0,u1,f0 are the initial data belong to a suitable space.The problem (1) is a general form of a model introduced by Kirchhoff [1]. To be more precise, Kirchhoff recommended a model denoted by the equation for f=g=0,
(2)ph∂2u∂t2+δ∂u∂t+g∂u∂t=ρ0+Eh2L∫0L∂u∂x2dx∂2u∂x2+fu,for0<x<L,t≥0, where ux,t is the lateral displacement, ρ is the mass density, h is the cross-section area, E is the Young modulus, L is the length, ρ0 is the initial axial tension, and f,g are the external forces. Furthermore, (2) is called a degenerate equation when ρ0=0 and nondegenerate one when ρ0>0.Time delays often appear in many various problems, such as thermal, economic phenomena, biological, chemical, and physical. Recently, the partial di/erential equations with time delay have become an active area, (see [2, 3] and references therein). Datko et al. [4] indicated that a small delay in a boundary control is a source of instability. An arbitrarily small delay may destabilize a system which is uniformly asymptotically stable without delay unless additional conditions or control terms have been used in many cases [5]. Additional control terms will be necessary to stabilize hyperbolic systems including delay terms, (see [6–8] and references therein). In [6], Nicaise and Pignotti studied the equation as follows:
(3)uttx,t−Δux,t+a0utx,t+autx,t−τ=0,where a0 and a are positive real parameters. The authors obtained that, under the condition 0≤α≤a0, the system is exponentially stable. In the case α≥α0, they obtained a sequence of delays that shows the solution is instable. In [8], Xu et al. obtained the same result similar to the [6] for the one space dimension by adopting the spectral analysis approach. In [9], Nicaise et al. studied the wave equation in one space dimension in the case of time-varying delay. In that work, the authors showed that an exponential stability result under the condition
(4)α≤1−dα0,where d is a constant such that
(5)τ′t≤d<1,∀t>0.In recent years, some other authors investigate hyperbolic type equation with delay term (see [10–16]).Without delay termμ2utx,t‐τr‐1utx,t−τ, in 2004, Li [17] studied the higher-order Kirchho/-type equation as follows:
(6)μtt+∫ΩDm/2u2dxq−Δmu+ututr=upu,where m>1 is a positive integer, and q,p,r>0 is a positive constant. The author obtained that the solution exists globally if p≤r, while if p>maxr,2q. He also established the blow-up result for E0<0. Later, in 2007, Messaoudi and Houari [18] obtained the blow-up of solutions with E0>0 of the equation (6). Then, Pişkin and Polat [19] considered global existence and decay estimates utilizing Nakao’s inequality of the equation (6).Without delay term, whenm=1 and q=0, equation (1) takes the form of a semilinear hyperbolic equation as follows:
(7)utt−Δu+ututr−1=up−1u.Georgiev and Todorova [20] obtained the blow-up of solutions for E0<0if1<r<p1<p<n/n−2forn≥3,p>1forn<3 of the equation (7). Under the condition of positive upper bounded initial energy, Vitillaro [21] proved the same results of equation (7). Also, Ohta [22, 23] studied related problems for the blow-up results of the equation (7).Messaoudi [24] studied the following equation
(8)utt+Δ2u+utru2t=upu2and obtained an existence result for the equation (8) and proved that the solution continues to exists globally if r≥p; however, if r<p and the initial energy is negative, the solution blows up in finite time. Chen [25] established that the solution of (8) blows up with E0>0. In the presence of strong damping term −Δut, Pişkin and Polat [26] obtained the decay estimates by using Nakao’s inequality of equation (8).Whenm=1 and without delay term, equation (1) takes the form the following Kirchhoff-type equation:
(9)utt‐∫ΩDu2dxγΔu+ututr=upu.Many authors had studied existence and blow-up results at night time for equation (9) (see [27–30]). Ono [30] proved the blow-up results if p>maxr,2γp<2/n−4forn≥5,p>0forn≤4 and E0<0 for equation (9). Later, Benaissa and Messaoudi [31] obtained the similar result for the generalized Kirchhoff-type equation as follows:
(10)utt−M∫Ωeϕx∇u2dxe−ϕxdiveϕx∇u+αutr−2ut=bup−2u,where M:R+⟶R+ and ϕx are bounded functions. Then, Wu [32], verified the same result of the general Kirchhoff-type equation
(11)utt−M∇u22Δu+utr−2ut=up−2u,with the positive upper bounded initial energy. In 2013, Ye [33] considered the global existence results by constructing a stable set in H01Ω and showed the decay by using a lemma of Komornik for the nonlinear Kirchhoff-type equation (11) with dissipative term. Moreover, Ye [34] obtained the global existence results by constructing a stable set in H0mΩ and showed the energy decay by using a lemma of V. Komornik for a nonlinear higher-order Kirchhoff-type equation with dissipative term is as follows:
(12)utt+A1/2u2pAu+autr−2ut=bup−2u,where A=−∆m,m>1 is a positive integer.Gao et al. [35] considered the Kirchhoff-type equation without delay term as follows:
(13)utt+MDmu22−Δmu+utr−2ut=up−2u.The authors obtained the blow-up of solutions forE0>0 under appropriate conditions for equation (13).In [36–40], some authors studied abstract evolution equations as follows:
(14)PuttAt,u+Qt,ut=Fuon suitable Banach space, and they proved some global nonexistence of solutions. Some other authors studied related problems (see [41–45]).Motivated by the above works, we deal with the existence, decay, and blow-up results for the higher-order Kirchhoff type equation (1) with delay term and source term. There is no research, to our best knowledge, related to the higher-order Kirchhoff-type ∫ΩAm/2u2dxqAmu equation (1) with delay u2utx,t−τr‐1utx,t−τ and source up−1u terms; hence, our work is the generalization of the above studies.This work consists of five sections in addition to the introduction: Firstly, in Sect.2, we recall some lemmas and assumptions. Then, in Section 3, we get the global existence of solutions. Moreover, in Section 4, we establish the decay results by using Nakao’s tecnique. Finally, in Section 5, we obtain the blow-up of solutions for negative initial energy.


## 2. Preliminaries

In this part, we present some lemmas and assumptions for the proof of our result. LetHmΩ denote the Sobolev space with the norm
(15)uHmΩ=∑α≤mDαuL2Ω21/2.,H0mΩ denotes the closure in HmΩ of C0∞Ω. For simplicity of notation, we denote by ⋅p the Lebesgue space LpΩ norm, ⋅ denotes L2Ω norm, and we write equivalent norm ∇⋅instead of H01Ω norm ⋅H01Ω. We denote by Cii=1,2,⋯,n various positive constants which may be different at different occurrences.Lemma 1 (see [46, 47] Sobolev-Poincarė inequality).
If2≤p≤2n/n−2m+2≤p<∞ifn=2m, then for some C∗,
up≤C∗−Δm/2u for u∈H0mΩ,
where we putα+=max0,α,1/α+=∞ if α+=0.Lemma 2 (see [48]).
Letϕt be nonincreasing and nonnegative function defined on 0,T,T>1 and satisfies
(16)ϕ1+αt≤w0ϕt−ϕt+1,t∈0,T
forw0 is a positive constant, and α is a nonnegative constant. Then, we have for each t∈0,T,
(17)ϕt≤ϕ0e−w1t−1+,α=0,ϕt≤ϕ0−α+w0−1αt−1+−1/α,α=0,
wheret−1+=maxt−1,0, and w1=lnw0/w0−1.
We make the assumptions on parametersr, p, and m as follows:
(A1)(18)1<p<∞,n≤2m,1<p<nn−2m,n>2m,
(A2)(19)1<r<∞,n≤2m,1<r<n+2mn−2m,n>2m.


## 3. Global Existence

In this part, we consider the global existence results of the problem (1). Firstly, we introduce the new function z similar to the [7],
(20)zx,k,t=utx,t−τk,x∈Ω,k∈0,1.Thus, we have(21)τztx,k,t+zkx,k,t=0,inΩ×0,1×0,∞.Hence, problem (1) can be transformed as follows:
(22)utt+∫ΩAm/2u2dxqAmu+μ1utx,tr−1utx,t+μ2zx,1,tr−1zx,1,t=up−1ux,t∈Ω×0,T,τxtx,k,t+zkx,k,t=0inΩ×0,1×0,∞,zx,k,0=f0x,−τkx∈Ω,ux,0=u0x,utx,0=u1xx∈Ω,∂iu∂vi=0,i=0,1,⋯,m−1x∈∂Ω.We define the energy functional for any regular solution of (22) as follows:
(23)Et=12ut2+12q+1Am/2u2q+1−1p+1up+1p+1+ςr+1∫Ω∫01zr+1x,k,sdkdx,such that
(24)τrμ2<ς<τr+1μ1−μ2.Also, have(25)Jt=Jut=12q+1Am/2u2q+1−1p+1up+1p+1+ςr+1∫Ω∫01zr+1x,k,sdkdx,(26)It=Iut=Am/2u2q+1−up+1p+1+ς∫Ω∫01zr+1x,k,sdkdx.We easily see that(27)Et=Jt+12ut2.Furthermore, we define(28)W=u:u∈H0mΩ∩H2mΩ,Iu>0∪0.Next, lemma gives that the energy functionalEt is a nonincreasing.Lemma 3.
Assume thatu,z is the solution of (22), then for t≥0,
(29)E′t=−μ1−ςτr−1−μ2r+1uttr+1r+1−ςτr+1−μ2rr+1∫Ωzr+1x,1,tdx≤0.Proof.
We multiply the first equation in (22) by ut, integrate over, and use integration by parts, and we obtain
(30)ddt12utt2+12q+1Am/2u2q+1−1p+1up+1p+1+μ1uttr+1r+1∫Ωμ2zx,1,tr−1zx,1,tutx,tdx=0.
Integrating (30) over 0,t, we get
(31)12utt2+12q+1Am/2u2q+1−1p+1up+1p+1+∫0tμ1ussr+1r+1ds+μ2∫0t∫Ωzx,1,sr−1zx,1,susx,sdxds=12u12.
We multiply the second equation in (22) by ςzr−1z and integrate the result over Ω×0,1×0,t, and we get
(32)ςr+1ddt∫0t∫Ω∫01zx,k,tr−1x,k,tztx,k,tdkdxds=−ςτr+1∫0t∫Ω∫01∂∂kzx,k,tr+1dkdxds=−ςτr+1∫0t∫Ωzx,1,tr+1−zx,0,tr+1dxds=−ςτr+1∫0t∫Ωzx,1,tr+1dxds+ςτr+1∫0tuttr+1r+1ds.
By combining (31) and (32), we arrive at
(33)Et+μ1−ςτr+1∫0tussr+1r+1ds+ςτr+1∫0t∫Ωzx,1,sr+1dxds+μ2∫0t∫Ωzx,1,sr−1zx,1,susx,sdxds=E0.
Utilizing the Young inequality on the fourth term of the left hand side of (33), we conclude that
(34)Et+μ1−ςτr+1−μ2r+1∫0tussr+1r+1ds+ςτr+1−μ2rr+1∫0t∫Ωzx,1,sr+1dxds=E0.
Deriving the (34), we have the desired result. Hence, the proof is completed.Remark 4.
From the condition (24), we obtain
(35)c1=μ1−ςτr+1−μ2r+1>0,c2=ςτr+1−μ2rr+1>0.Lemma 5.
Assume that (19) and p>2q+1 hold. Let u0∈W and u1∈H0mΩ, such that
(36)β=C∗2q+1p+1p−2q−1E0p−2q−1/2q+1<1,then u∈W for each t≥0.Proof.
It follows the continuity ofut, since I0>0, such that
(37)It>0,
for some interval neart=0. Assume that Tm>0 is a maximal time, when (26) holds on 0,Tm.
By (25) and (26), we obtain
(38)Jt=1p+1It+p−2q−12q+1p+1Am/2u2q+1+p−rr+1p+1ς∫Ω∫01zr+1x,k,tdkdx≥p−2q−12q+1p+1Am/2u2q+1.
From (23), (38), and Lemma 3, we have
(39)Am/2u2q+1≤2q+1p+1p−2q−1Jt≤2q+1p+1p−2q−1Et≤2q+1p+1p−2q−1E0.
Using Lemma1 and (39), we get
(40)up+1p+1≤C∗Am/2up+1=C∗Am/2up−2q−1Am/2u2q+1≤C∗2q+1p+1p−2q−1E0p−2q−1/2q+1Am/2u2q+1=βAm/2u2q+1<Am/2u2q+1ont∈0,Tm.
Thus, from (26), we arrive at It>0 for all t∈0,Tm. Tm is extended to T, by repeating the procedure. Hence, the proof is completed.Lemma 6.
Suppose that the assumptions of Lemma5 hold. Then, there exists η1=1−β, such that
(41)up+1p+1≤1−η1Am/2u2q+1.Proof.
By (40), we obtain
(42)up+1p+1≤βAm/2u2q+1.
Letη1=1−β; therefore, we obtain the result.Remark 7.
By Lemma6, we conclude that
(43)Am/2u2q+1≤1η1It.Theorem 8.
Assume that the assumptions (A2),μ2<μ1, and p>2q+1 hold. Let u0∈W satisfying (36) and f0∈L2Ω×0,1 be given. Then, the solution of problem (22) is global.Proof.
It is sufficient to show thatut2+Am/2u2q+1 is bounded independently of t. To obtain this, by using (23) and (26), we have
(44)E0≥Et=12ut2+12q+1Am/2u2q+1−1p+1up+1p+1+ςr+1∫Ω∫01zr+1x,k,xdkdx=12ut2+p−2q−12q+1p+1Am/2u2q+1+p−rr+1p+1ς∫Ω∫01zr+1x,k,tdkdx+1p+1It≥12ut2+p−2q−12q+1p+1Am/2u2q+1,since It≥0. Thus,
(45)ut2+Am/2u2q+1≤CE0,where C=max2,2q+1p+1/p−2q−1. Therefore, we obtain the global existence of solutions. Therefore, we completed the proof.


## 4. Decay of Solution

In this part, we obtain the decay of solutions of the problem (22) by using Nakao’s technique.Theorem 9.
Assume that the assumption (A2) and (36) hold. Let u0∈W, f0∈L2Ω×0,1, be given. Hence, we have following decay estimates:
(46)Et≤E0e−w1t−1+,ifr=1,E0−α+C7−1αt−1+−1/α,ifr>1,where w1,α and C7 are positive constants which will be defined later.Proof.
We integrate (29) over t,t+1,t>0, to get
(47)Et−Et+1=Dtr+1,where
(48)Dtr+1=c1∫tt+1utr+1r+1ds+c2∫tt+1∫Ωzr+1x,1,sdxds.
From (48) and Hölder inequality, we see that
(49)∫tt+1∫Ωut2dxdt+∫tt+1∫Ωzx,1,s2dxds≤xΩDt2,
wherecΩ=volΩ.Therefore, by (49), there exists t1∈t,t+1/4 and t2∈t+3/4,t+1, so that
(50)utti2+zx,1,ti2≤cΩDt2,i=1,2.
We multiply the first equation in (22) by u and integrate over Ω×t1,t2. Use integration by parts, Hölder’s inequality, adding, and subtracting the term ∫t1t2∫Ω∫01ςzr+1x,k,tdkdxdt, we have
(51)∫t1t2Itdt≤utt12ut12+utt22ut22+∫t1t2ut2dt+∫t1t2∫01ς∫Ωzr+1x,k,tdxdkdt−μ1∫t1t2∫Ωutr−1utudxdt−μ2∫t1t2∫Ωzx,1,tr−1zx,1,tudxdt.
Now, we estimate the right hand side for (51).
From (39), (50), and Lemma 1, we obtain
(52)utti2uti2≤C1Dtsupt1≤s≤t2E1/2s,where C1=2C∗2q+1p+1/p−2q−1E01/2q+1.
By using (32) that
(53)∫t1t2∫01∫Ωzr+1x,k,tdxdkdt≤12τ∫t1t2∫t1t2utsr+1r+1dsdυ≤∫t1t2dυ∫t1t2utsr+1r+1ds≤t2−t1Dtr+1.
Utilizing Hölder inequality, we get(54)∫t1t2∫Ωutr−1utudxdt≤∫t1t2uttr+1rutr+1.
Utilizing the Sobolev-Poincare inequality and (39), we have
(55)∫t1t2uttr+1rutr+1dt≤C∗∫t1t2uttr+1rAm/2udt≤C∗2q+1p+1p−2q−1E01/2q+1∫t1t2uttr+1rE1/2sdt≤C∗2q+1p+1p−2q−1E01/2q+1supt1,≤s≤t2E1/2s∫t1t2uttr+1rdt≤C∗2q+1p+1p−2q−1E01/2q+1supt1,≤s≤t2E1/2sDtr.
Also, we obtain(56)∫t1t2∫Ωzx,1,tr−1zx,1,tudxdt≤C∗∫t1t2zx,1,tr+1rur+1dt≤C∗2q+1p+1p−2q−1E01/2q+1∫t1t2zx,1,tr+1rE1/2sdt≤C∗2q+1p+1p−2q−1E01/2q+1supt1≤s≤t2E1/2s∫t1t2zx,1,tr+1rdt≤C∗2q+1p+1p−2q−1E01/2q+1supt1≤s≤t2E1/2sDtr.
Then, from (51)-(56), we get
(57)∫t1t2Itdt≤C2supt1≤s≤t2E1/2sDt+Dt2+t2−t1Dtr+1+2C∗2q+1p+1p−2q−1E01/2q+1supt1≤s≤t2E1/2sDtr.
Moreover, by (23), (26), and Remark 7, we have
(58)Et≤12ut2+C3It,
whereC3=1/η1p−2q−1/2q+1p+1+1/p+1.
Integrating (58) over t1,t2, we get
(59)∫t1t2Etdt≤12∫t1t2ut2dt+C3∫t1t2Itdt.
Hence, from (57) and (59), we obtain
(60)∫t1t2Etdt≤12CDt2+C3C2supt1≤s≤t2E1/2sDt+Dt2+t2−t1Dtr+1+2C∗2q+1p+1p−2q−1E01/2q+1supt1≤s≤t2E1/2sDtr.
Integratingd/dtEt over t,t2, we conclude that
(61)Et=Et2+∫tt2μ1−ςτr+1−μ2r+1ussr+1r+1ds+∫tt2ςτr+1−μ2rr+1∫Ωzx,1,sr+1dxds.
Thus, sincet2−t1≥1/2, we arrive at
(62)∫t1t2Etdt≥t2−t1Et2≥12Et2.
Hence,(63)Et2≤2∫t1t2Etdt.
As a result, from (47), (60), (61), (63), and since t1,t2∈t,t+1, we get
(64)Et≤2∫t1t2Etdt+∫tt+1μ1−ςτr+1−μ2r+1ussr+1r+1ds+∫tt+1ςτr+1−μ2rr+1∫Ωzx,1,sr+1dxds=2∫t1t2Etdt+Dtr+1.
Then, from (60), we have
(65)Et≤12C+C3C2Dt2+C3C2Dtr+1+C4Dt+DtrE1/2t.
Thus, utilizing Young inequality, we have(66)Et≤C5Dt2+Dtr+1+dt2r.Hence, we have the decay estimates as follows:Case 1.
Ifr=1, by (66), we obtain
(67)Et≤3C5Dt2=3C5Et−Et−1.
Utilizing Lemma2, we have
(68)Et≤E0e−w1t−1+,where w1=ln3C5/3C5−1.Case 2.
Ifr>1, by (66), we have
(69)Et≤C5Dt21+Dtr−1+Dt2r−2.
Then, by (47), since Et≤E0,∀t≥0, we see that
(70)Et≤C51+Er−1/r+10+E2r−1/r+10Dt2≤C6Dt2,t≥0.
Then, we get(71)Etr+1/2≤C7Dtr+1≤C7Et−Et+1.
Therefore, by (71) and Lemma 2, we obtain
(72)Et≤E0−α+C7−1αt−1+−1/α.
Thus, we completed the proof of Theorem9.


## 5. Blow-Up of Solution

In this part, we get the blow-up of solutions for negative initial energy, in the caser>1.Theorem 10.
Letu0,u1∈H2mΩ∩H0mΩ×H0mΩ and f0∈L2Ω×0,1 be given. Assume that p>max2,r,2q+1 and the assumptions (A1)-(A2) hold. Then, the solution of (22) blows up in a finite time with E0<0.Proof.
Setting(73)Ht=−Et,
from Lemma3, we obtain
(74)H′t≥−μ1−ςτr+1−μ2r+1uttr+1r+1−ςτr+1−μ2rr+1∫Ωzr+1x,1,tdx.
Thus,(75)0<H0≤Ht≤1p+1up+1p+1,t>0.
Let(76)Mt=u22.
Differentiating (76) twice, we get
(77)M′t=2∫Ωutudx,M″t=2ut2+2∫Ωuttudx.
Using the first equation in (22), to have
(78)M″t=2ut2−2Am2u22q+1−2μ1∫Ωutx,tr−1uutx,tdx−2μ2∫Ωzx,1,tr−1uzx,1,tdx+2up+1p+1,
we add and subtract the term2p+1Ht, and then (78) becomes the form
(79)M″t≥p+3ut2+2p+1Ht+p+1q+1−2Am2u22q+1−2μ1∫Ωutx,tr−1uutx,tdx−2μ2∫Ωzx,1,tr−1uzx,1,tdx+2ςp+1r+1∫Ω∫01zr+1x,k,sdkdx.
Now, we define(80)Lt=Ht1−κ+2ϵM′t.
Differentiating (80), we obtain
(81)L′t=1−κHt−κH′t+2εM″t.
Replacing (79) in (81), we arrive at
(82)L′t≥1−κHt−κH′t2εp+3μt2+4εp+1Ht+2εp+1q+1−2Am2μ22q+1−4εμ1∫Ωμtx,tr−1uutx,tdx−4εμ2∫Ωzx,1,tr−1uzx,1,tdx+4εςp+1r+1∫Ω∫01zr+1x,k,sdkdx.
From (75) and utilizing Hölder inequality, we get
(83)∫Ωutx,tr−1uutx,tdx≤utr+1rur+1≤c1ur+1r+1/p+1ur+11−r+1/p+1ur+1r≤c2ur+1r+1/p+1Ht1p+1−r+1p+12utr+1r.
From Young’s inequality and (74), we have
(84)∫Ωutx,tr−1uutx,tdx≤c3ρ1/1+pμr+1r+1H0−k¯+ρ−r′H0k−k¯H′tHt−k,where k¯=1/p+1−r+1/p+12>0,ρ>0,r′=r+1/r, letting 0<k<k¯. In a similar way, we obtain
(85)∫Ωuzx,1,tr−1zx,1,tdx≤c3ρ11+pμr+1r+1H0−k¯+ρ−r′H0k−k¯H′tHt−k.
By using (82), (84), and (85), to have
(86)L′t≥1−κ−4εμ1+μ2Hk−k¯0ρ−r′Ht−κH′t−4εμ1+μ2H0−k¯ρ1/1+pur+1r+1+2εp+3ut2+4εp+1Ht+2εp+1q+1−2Am/2μ22q+1+4εςp+1r+1∫Ω∫01zr+1x,k,sdkdx,for ε sufficiently small, we obtain
(87)1−κ−4εμ1+μ2Hk−k¯0ρ−r′≥0.
Settings=r+1≤p+1 such that
(88)μr+1s≤c1Am/2μ+μp+1p+1,where c=4μ1+μ2H0−k¯ρ1/p+1c1 and taking p+1/q+1−2>c. Thus, we have
(89)L′t≥2εp+1q+1−2−cAm/2u22q+1−εcμp+1p+1+4εp+1Ht+2εp+3μ2+4εζp+1r+1∫Ω∫01zr+1x,k,sdkdx.
By using the notationsa1=2p+1/q+1−2−c, a2=c, a3=4p+1, and a4=2p+3, (89) takes the form
(90)L′t≥a1εAm/2μ22q+1−εa2μp+1p+1+εa3Ht+εa4μt2.
Similarly to the approach of Messaoudi [49], we assume that p=2a5+p−2a5, where a5<mina1,a2,a3,a4, and then (90) becomes the form
(91)L′t≥a1−a5εAm/2μ22q+1+εa5−a2μp+1p+1+εa3−a5Ht+εa4−a5μt2.Then,
(92)L′t≥δεAm2u22q+1+μtp+1p+1+Ht+ut2.We conclude that
(93)L′t≥δεup+1p+1+Ht+ut2,where δ>0 is the minimum of the coefficients of up+1p+1,Ht,ut2. Pick out ε such that
(94)L0=H1−κ0+2ε∫Ωu1u0dx>0.
As a result, we getsettingϖ=1/1−k, and since k<k¯<1, we see that 1<ϖ<1/1−k¯. Set
(95)Lt=Ht1−κ+2ε∫Ωuutdx.Then,
(96)Lt=Ht1/ϖ+2ε∫Ωuutdx≤Ht1/ϖ+2ε∫Ωuutdx+2E1up+1p+1/ϖ.
Utilizing Young, Hölder’s inequalities, and (96), we conclude that
(97)Ltϖ≤Ht1/ϖ+2ε∫Ωuutdx+2E1up+1p+1/ϖϖ≤2ϖ−1Ht+2ε∫Ωuutdx+2E1up+1p+1/ϖϖ≤2ϖ−1Ht+2ϖ−12ε∫Ωuutdxϖ+2E1up+1p+1≤2ϖ−1Ht+2ϖ−1βϖut2ϖu2ϖ+2E1up+1p+1≤2ϖ−1Ht+2ϖ−1βϖut2ϖu2ϖ+2ϖ−12E1up+1p+1≤c2Ht+ut2ϖu2ϖ+up+1p+1,where c2=max2ϖ−1,βϖ. Furthermore, for p>1, utilizing Hölder and Young inequalities, we obtain
(98)ut2ϖu2ϖ≤c3ut2ϖup+1ϖ≤c4ut22+up+121−k/1−2k,up+121−k/1−2k=up+1p+1up+121−k/1−2k−p+1≤c5H021−k/1−2kp+1−1up+1p+1.
Then, (97) becomes the form
(99)Ltϖ≤c6Ht+ut2+up+1p+1.
By combining (93) and (99), we conclude that
(100)L′t≤c7Ltϖ,c7>0,ϖ>1.
Therefore, a simple integration over (0,t), we have the desired result. Hence, we completed the proof.


## 6. Conclusions

Time delays often appear in many various problems, such as, thermal, economic phenomena, biological, chemical, and physical. Recently, the partial differential equations with time delay have become an active area (see [2, 3] and references therein). In recent years, there has been published much work concerning the wave equation with constant delay or time-varying delay. However, to the best of our knowledge, there were no global existence, decay, and blow-up results for the higher-order Kirchhoff-type equation with delay term. Firstly, we have been obtained the global existence result. Later, we have been established the decay results by using Nakao’s technique. Finally, we have proved the blow-up of solutions with negative initial energy for the problem (1) under the sufficient conditions in a bounded domain. In the next work, we will extend our current study to more general case of the problem (1).



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*Source: 4414545-2021-09-02.xml*