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 : yonghann@cu.ac.kr
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
e-mail : jyepark@pusan.ac.kr
* 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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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:

(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

f(x,u)μuα-1+d1up-1

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

f(x,u)u-(p-ɛ)Φ(x,u)d2up

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

Q(x,t,v)v0Q(x,t,v)[d(x,t)]1/m[Q(x,t,v)v]1/m

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

d(x,t)0,d(·,t)p/(p-m)Lloc(J).

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

Z=L([0,T];WΓ01,α(Ω))W1,([0,T];L2(Ω))W1,β([0,T];WΓ01,β(Ω))W1,m([0,T];Lm(Ω)),

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

E(t)=l-1lΩut(x,t)ldx+1αΩa(x)u(x,t)αdx-ΩΦ(x,u(x,t))dx+ξ2Ω01z2(x,ρ,t)dρdx+12Γ1h(x)q(x)y2(x,t)dΓ,

where

τμ2<ξ<τ(2μ1-μ2),μ1>μ2.

We set

λ1=(A0-μμ0)1/(p-α)(d1B1p)-1/(p-α),E1=(1α-1p)(a0-μμ0)p/(p-α)(d1B1p)-α/(p-α),

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

B1-1=inf{uα:uW01,α(Ω),up=1}.

We also set

Σ={(λ,E)2λ>λ1,E<E1}.
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

(uα,E(0))Σ.

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,

ddtE(t)(l-1)Ωut(x,t)l-2ut(t)utt(t)dx+Ωa(x)u(t)α-2u(t)ut(t)dx-ΩQ(x,t,ut(t))ut(t)dx-c0Ω(ut2(t)+z(x,1,t))dx-Γ1h(x)k(x)yt2(t)dΓ0.
Proof

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

ddt{l-1lΩut(t)ldx+1αΩa(x)u(t)αdx-ΩΦ(x,u(t))dx}-Γ1h(x)yt(t)ut(t)dΓ=(l-1)Ωut(t)l-2ut(t)utt(t)dx+Ωa(x)u(t)α-2u(t)ut(t)dx-Ωut(t)dx-ΩuΦ(x,u)ut(t)dx-μ1Ωut2(t)dx-μ2Ωz(x,1,t)ut(t)dx.

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

-Γ1h(x)yt(t)ut(t)dΓ=Γ1h(x)k(x)yt2(t)dΓ+Γ1h(x)q(x)y(t)yt(t)dΓ.

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

ddt{ξ2Ω01z2(x,ρ,t)dρdx}=-ξτΩ01z(x,ρ,t)zρ(x,ρ,t)dρdx=-ξ2τΩ01ρz2(x,ρ,t)dρdx=ξ2τΩ[z2(x,0,t)-z2(x,1,t)]dx=ξ2τ[Ωut2(t)dx-Ωz2(x,1,t)dx],

and

-μ2Ωz(x,1,t)ut(t)dxμ22[Ωut2(t)dx+Ωz2(x,1,t)dx].

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

ddtE(t)-Ωut(t)2dx-Ωut(t)βdx-ΩQ(x,t,ut(t))ut(t)dx-(μ1-ξ2τ-μ22)Ωut2(t)dx-(ξ2τ-μ22)Ωz2(x,1,t)dx-Γ1h(x)k(x)yt2(t)dΓ.

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

ddtE(t)-Ωut(t)2dx-Ωut(t)βdx-ΩQ(x,t,ut(t))ut(t)dx-c0Ω[ut2(t)+z2(x,1,t)]dx-Γ1h(x)k(x)yt2(t)dΓ0.

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).
Proof

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

E(t)1αΩa(x)u(t)αdx-ΩΦ(x,u(t))dx.

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

Φ(x,u(t))=01f(x,τu(t))u(t)dτμαu(t)α+d1pu(t)p,

and then

ΩΦ(x,u(t))dx=01f(x,τu(t))u(t)dτμαu(t)αα+d1pu(t)pp,

Therefore

E(t)a0αu(t)αα-μαu(t)αα-d1pu(t)pp(a0-μμ0)1αu(t)αα-d1pu(t)pp(a0-μμ0)1αu(t)αα-d1B1p1pu(t)αp=(a0-μμ0)1αλα-d1B1p1pλp:=ϕ(λ),

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

E(t0)ϕ(u(t0)α)>ϕ(λ2)=E(0).

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

d1pu(t)pp(a0-μμ0)1αu(t)αα-E(t)(a0-μμ0)1αu(t)αα-E0(a0-μμ0)1αλ2α-ϕ(λ2)=d1B1p1pλ2p.

Thus, the proof is complete.

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

H(t)=E1-E(t),t0.

Then we have the following Lemma.

Lemma 3.3

For all tJ, we have

0<H(0)H(t)d1pu(t)pp.
Proof

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

H(t)H(0)=E1-E(0)>0,t0.

From (3.12), we obtain

H(t)=E1-E(t)ϕ(λ1)-(a0-μμ0)1αu(t)αα+d1pu(t)pp=(a0-μμ0)1α(λ1α-u(t)αα)-d1B1p1pλ1p+d1pu(t)pp.

From (3.8), ||∇u(t)||α> λ1, we get

H(t)d1pu(t)pp.

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

Now, we define

L(t)=H1-σ(t)+ɛΩu(t)ut(t)l-2ut(t)dx+μ1ɛ2Ωut2(t)dx-ɛ2Γ1h(x)k(x)y2(t)dΓ-ɛΓ1h(x)u(t)y(t)dΓ,

for ɛ small to be chosen later and

0<σmin{α-2p,α-βp(β-1),α-mp(m-1),α-lαl,kɛα-1}.

By taking a derivative of (3.17) we have

L(t)=(1-σ)H-σ(t)H(t)+ɛut(t)ll+ɛΩu(t)(ut(t)l-2ut(t))tdx+μ1ɛΩu(t)ut(t)dx-ɛΓ1h(x)k(x)y(t)yt(t)dΓ-ɛΓ1h(t)ut(t)y(t)dΓ-ɛΓ1h(x)u(t)yt(t)dΓ.

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

L(t)=(1-σ)H-σ(t)H(t)+ɛut(t)ll+ɛΩ(Δut(t)+div(a(x)u(t)α-2u(t))+div(ut(t)β-2ut(t))-Q(x,t,ut)-μ1ut(t)-μ2z(x,1,t)+f(x,u(t)))u(t)dx+μ1ɛΩu(t)ut(t)dx-ɛΓ1h(x)k(x)y(t)yt(t)dΓ-ɛΓ1h(t)ut(t)y(t)dΓ-ɛΓ1h(x)u(t)yt(t)dΓ=(1-σ)H-σ(t)H(t)+ɛut(t)ll-ɛΩut(t)u(t)dx-ɛΩa(x)u(x,t)αdx-ɛΩut(t)β-2ut(t)u(t)dx+ɛΓ1(ut(t)ν+u(t)α-2u(t)ν+ut(x,t)β-2ut(x,t)ν)u(t)dΓ-ɛΩQ(x,t,ut)u(t)dx+ɛΩf(x,u(x,t))u(t)dx-μ1ɛΩu(t)ut(t)dx-μ2ɛΩz(x,1,t)u(t)dx+μ1ɛΩu(t)ut(t)dx-ɛΓ1h(x)k(x)y(t)yt(t)dΓ-ɛΓ1h(x)ut(t)y(t)dΓ-ɛΓ1h(x)u(t)yt(t)dΓ=(1-σ)H-σ(t)H(t)+ɛut(t)ll-ɛΩut(t)u(t)dx-ɛΩa(x)u(x,t)αdx-ɛΩut(t)β-2ut(t)ut(t)dx-ɛΩQ(x,t,ut)u(t)dx+ɛΩf(x,u(x,t))u(t)dx-μ2ɛΩz(x,1,t)u(t)dx+ɛΓ1h(x)q(x)y2(t)dΓ

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

ΩQ(x,t,u)u(t)dxΩu(t)[d(x,t)]1/m[Q(x,t,ut(t))ut(t)]1/mdxδmmΩu(t)md(x,t)dx+m-1mδ-mm-1ΩQ(x,t,ut(t))ut(t)dxδmmu(t)pmd(t)p/(p-m)+m-1mδ-mm-1ΩQ(x,t,ut(t))ut(t)dxδmCmu(t)pm+m-1mδ-mm-1ΩQ(x,t,ut(t))ut(t)dx.

By Young’s inequality, we get

Ωut(t)ut(t)dx14μΩu(t)2dx+μΩut(t)2dx,Ωut(t)β-2ut(t)u(t)dxηββΩu(t)βdx+β-1βη-ββ-1Ωut(t)βdx,μ2Ωu(t)z(x,1,t)dxμ24ρΩu2(t)dx+μ2ρΩz2(x,1,t)dx.

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

L(t)(1-σ)H-σ(t)H(t)+ɛut(t)ll-ɛ4μΩu(t)2dx-ɛμΩut(t)2dx-ɛΩa(x)u(t)αdx-ɛηββΩu(t)βdx-ɛ(β-1)βη-ββ-1Ωut(t)βdx-ɛδmCmu(t)pm-ɛ(m-1)mδ-mm-1ΩQ(x,t,ut(t))ut(t)dx-ɛμ24ρΩu2(t)dx-ɛμ2ρΩz2(x,1,t)dx+ɛΩf(x,u(x,t))u(t)dx+ɛΓ1h(x)q(x)y2(t)dΓ

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

δ-mm-1=M1H-σ(t),μ=M2H-σ(t)η-ββ-1=M3H-σ(t),ρ=M4H-σ(t),

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

L(t)(1-σ)H-σ(t)H(t)+ɛut(t)ll-ɛ4M2Hσ(t)Ωu(t)2dx-ɛΩa(x)u(t)αdx-ɛM3-(β-1)βHσ(β-1)(t)Ωu(t)βdx-ɛM1(m-1)CmHσ/(m-1)(t)u(t)pm-ɛμ24M4Hσ(t)Ωu2(t)dx-ɛ[M2Ωut(t)2dx+(β-1)βM3Ωut(t)βdx+(m-1)mM1ΩQ(x,t,ut(t))ut(t)dx+μ2M4Ωz2(x,1,t)dx]H-σ(t)+ɛΩf(x,u(x,t))u(t)dx+ɛΓ1h(x)q(x)y2(t)dΓ.

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

L(t)(1-σ-ɛM)H-σ(t)H(t)+ɛut(t)ll-ɛ4M2Hσ(t)Ωu(t)2dx-ɛΩa(x)u(t)αdx-ɛM3-(β-1)βHσ(β-1)(t)Ωu(t)βdx-ɛCmM1(m-1)Hσ/(m-1)(t)u(t)pm-ɛμ24M4Hσ(t)Ωu2(t)dx+ɛMH-σ(t)Γ1h(x)k(x)yt2(t)dΓ+ɛΩf(x,u(t))u(t)dx+ɛΓ1h(x)q(x)y2(t)dΓ.

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

zδ(1+1/a)(z+a),z>0,0<δ1,a>0,

we have (see[15])

Hσ(t)Ωut(t)2dxc(Ω)(B1pd1p)σ(Ωu(t)αdx)(pσ+2)/αd(B1pd1p)σ(Ωu(t)αdx+H(t)),Hσ(β-1)(t)Ωut(t)βdxc(Ω)(B1pd1p)σ(β-1)(Ωu(t)αdx)(pσ(β-1)+β)/αd(B1pd1p)σ(β-1)(Ωu(t)αdx+H(t)),Hσ(m-1)u(t)pmc(Ω)(B1pd1p)σ(m-1)B1m(Ωu(t)αdx)(pσ(m-1)+m)/αd(B1pd1p)σ(m-1)B1m(Ωu(t)αdx+H(t)),

and

Hσ(t)Ωut(t)2dxc(Ω)(B1pd1p)σB12(Ωu(t)αdx)(σp)/αd(B1pd1p)σB12(Ωu(t)αdx+H(t)),

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

L(t)(1-σ)-ɛM)H-σ(t)H(t)+kH(t)+(ɛ+k(l-1)l)ut(t)ll-ɛc2M2(Ωu(t)αdx+H(t))-ɛΩa(x)u(t)αdx-ɛc3M3β-1(Ωu(t)αdx+H(t))+kαΩa(x)u(t)αdx-ɛc1M1m-1(Ωu(t)αdx+H(t))-ɛc4M4(Ωu(t)αdx+H(t))+ɛΩf(x,u(t))u(t)dx+ɛΓ1h(x)q(x)y2(t)dΓ-kΩΦ(x,u(t))dx+kξ2Ω01z2(x,ρ,t)dρdx+k2Γ1h(x)q(x)y2(t)dΓ-kE1+ɛMH-σ(t)Γ1h(x)k(x)yt2(t)dΓ,

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

-kE1-kE1B1-pλ1-pu(t)pp=-kd1(1α-1p)u(t)pp.

From (2.2), we can choose k satisfying

αɛk<pɛmin {αd2(p-α)d1,1}

and

ɛΩf(x,u(t))u(t)dx-kΩΦ(x,u(t))dx-kE1ɛd2u(t)pp-kd1(1α-1p)u(t)pp0.

Thus, it follows that

L(t)(1-σ)-ɛM)H-σ(t)H(t)+(ɛ+k(l-1)l)ut(t)llɛ(kɛ-c2M2-c3M3β-1-c1M1m-1-c4M4)H(t)+ɛ((kɛα-1)a0-c2M2-c3M3β-1-c1M1m-1-c4M4)Ωu(t)αdx+ɛΓ1h(x)q(x)y2(t)dΓ+kξ2Ω01z(x,ρ,t)dρdx+k2Γ1h(x)q(x)y2(t)dΓ+ɛMH-σ(t)Γ1h(x)k(x)yt2(t)dΓ.

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

ɛMH-σ(t)Γ1h(x)k(x)yt2(t)dΓ0,

we deduce

L(t)(1-σ)-ɛM)H-σ(t)H(t)+γɛ(H(t)+ut(t)ll+Ωut(t)αdx+Γ1h(x)q(x)y2(t)dΓ+Ω01z2(x,ρ,t)dρdx),

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

L(0)=H1-σ(0)+ɛΩu0u1l-2u1dx+μ1ɛ2Ωu02dx-ɛ2Γ1h(x)k(x)y02dΓ-ɛΓ1h(x)u0y0dΓ>0.

Then from(3.33) we get

L(t)L(0)0,         t0,

and

L(t)γɛ(H(t)+ut(t)ll+Ωu(t)αdx+Γ1h(x)q(x)y2(t)dΓ+Ω01z2(x,ρ,t)dρdx).

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

L(t)H1-σ(t)+ɛΩu(t)ut(t)l-2ut(t)dx+μ1ɛ2Ωu2(t)dx-ɛΓ1h(x)u(t)y(t)dΓ.

Then the above inequality leads to

L11-σ(t)[H1-σ(t)+ɛΩu(t)ut(t)l-2ut(t)dx+μ1ɛ2Ωu2(t)dx-ɛΓ1h(x)u(t)y(t)dΓ]1/(1-σ)C(ɛ,μ1,σ)[H(t)+Ωu(t)ut(t)l-2ut(t)dx11-σ+(Ωu2(t)dx)11-σ+Γ1h(x)u(t)y(t)dΓ11-σ].

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

|Ωu(t)ut(t)|l-2ut(t)dx(Ωu(t)ldx)1/l(Ωut(t)ldx)(l-1)/l(Ωu(t)σdx)1/α(Ωut(t)ldx)(l-1)/lc[(Ωu(t)αdx)l(1-σ)/[l(1-σ)-(l-1)]α+(Ωut(t)ldx)(l-σ)].

From (3.18) and (3.29), we obtain

|Ωu(t)|ut(t)l-2ut(t)dx1/(1-σ)c[(Ωu(t)αdx)l/[l(1-σ)-(l-1)]α+Ωut(t)ldx]c[(1+1H(0))(Ωu(t)αdx+H(t))+Ωut(t)ldx].

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

|Ωu(t)ut(t)l-2ut(t)dx|1/(1-σ)C[H(t))+u(t)αα+ut(t)ll].

Furthermore, by the same method, we deduce

Γ1h(x)u(t)y(t)dΓ=|Γ1h(x)q(x)q(x)u(t)y(t)dΓ|h12q12q0(Γ1h(x)q(x)y2(t)dΓ)12(Γ1u2(t)dΓ)12.

Similarly, we find

Γ1h(x)u(t)y(t)dΓ=|Γ1h(x)q(x)q(x)u(t)y(t)dΓ|h12q12q0(Γ1h(x)q(x)y2(t)dΓ)12(Γ1u2(t)dΓ)12.

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

Γ1h(x)u(x)y(t)dΓc5h12q12q0(Γ1h(x)q(x)y2(t)dΓ)12(Ωu(t)αdx)1α.

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

(Γ1h(x)u(t)y(t)dΓ)11-σc6(Γ1h(x)q(x)y2(t)dΓ)12(1-σ)(Ωu(t)αdx)1α(1-σ).

Using Young’s inequality, we write

(Γ1h(x)u(t)y(t)dΓ)11-σc7[(Ωu(t)αdx)2α(1-2σ)+Γ1h(x)q(x)y2(t)dΓ],

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

(Γ1h(x)u(t)y(t)dΓ)11-σc8[H(t)+u(t)αα+Γ1h(x)q(x)y2(t)dΓ],

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

L11-σ(t)C*[H(t)+ut(t)ll+u(t)αα+Γ1h(x)q(x)y2(t)dΓ++Ω01z2(x,ρ,t)dρdx],t0,

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

L(t)ξL11-σ(t),t0.

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

Lσ1-σ(t)1L-σ1-σ(0)-ξσ1-σt,t0.

Therefore L(t) blow up in finite time

TT*=1-σξσLσ1-σ(0).

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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