# A New Result for a Blow-up of Solutions to a Logarithmic Flexible Structure with Second Sound

Authors: Ahlem Merah; Fatiha Mesloub; Salah Mahmoud Boulaaras; Bahri-Belkacem Cherif
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

This paper is concerned with a problem of a logarithmic nonuniform flexible structure with time delay, where the heat flux is given by Cattaneo’s law. We show that the energy of any weak solution blows up infinite time if the initial energy is negative.


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## Body

## 1. Introduction

In this work, we consider the vibrations of an inhomogeneous flexible structure system with a constant internal delay and logarithmic nonlinear source term:(1)mxutt−pxux+2δxutxx+ηθx+μutx,t−τ0=uup−2lnuγ,x∈0,L,t>0,θt+kqx+ηutx=0x∈0,L,t>0,τqt+βq+kθx=0x∈0,L,t>0,with boundary conditions(2)u0,t=uL,t=0;θ0,t=θL,t=0,t≥0,and initial conditions(3)ux,0=u0x,utx,0=u1x;θx,0=θ0x;qx,0=q0x,x∈0,L,where ux,t is the displacement of a particle at position x∈0,L, and the time t>0. η>0 is the coupling constant depending on the heating effect,p≥2,γ,β, and k are positive constants, and μ is a real number. τ>0 is the relaxation time describing the time lag in the response for the temperature, and τ0>0 represents the time delay in particular if τ=01.1 reduces to the viscothermoelastic system with delay, in which the heat flux is given by Fourier’s law instead of Cattaneo’s law, where q=qx,t is the heat flux, and mx,δx, and px are responsible for the inhomogeneous structure of the beam and, respectively, denote mass per unit length of structure, coefficient of internal material damping (viscoelastic property), and a positive function related to the stress acting on the body at a point x. The model of heat condition, originally due to Cattaneo, is of hyperbolic type. We recall the assumptions of mx,δx, and px in [1, 2] such that(4)m,δ,p∈W1,∞0,L,mx,δxandpx>0,∀x∈0,L.In these kinds of problems, Gorain [3] in 2013 has established uniform exponential stability of the problem(5)mxutt−pxux+2δxutxx=fx,on0,L×ℝ+,which describes the vibrations of an inhomogeneous flexible structure with an exterior disturbing force. More recently, Misra et al. [4] showed the exponential stability of the vibrations of a inhomogeneous flexible structure with thermal effect governed by the Fourier law.(6)mxutt−pxux+2δxutxx−kθx=fx,θt−θxx−kuxt=0.In addition, we can cite other works in the same form like the system in [5]; Racke studied the exponential stability in linear and nonlinear 1d of thermoelasticity system with second sound given by(7)mxutt−pxux+2δxutxx−kθx=0,on0,L×ℝ+θt+kqx+ηutx=0,on0,L×ℝ+τqt+βq+kθx=0,on0,L×ℝ+,Now for the multidimensional system, Messaoudi in [6] established a local existence and a blow-up result for a multidimensional nonlinear system of thermoelasticity with second sound (see in this regard Refs. [7–10]); for the same problem above, Alves et al. proved that system (7) is polynomial decay (see [1]), with boundary and initial conditions:(8)u0,t=uL,t=0;θ0,t=θL,t=0,t≥0,ux,0=u0x,utx,0=u1x;θx,0=θ0x;qx,0=q0x,x∈0,L.We know that the dynamic systems with delay terms have become a significant examination subject in differential condition since the1970s of the only remaining century. The delay effect that is similar to memory processes is important in the research of applied mathematics such as physics, noninstant transmission phenomena, and biological motivation; model (7) is related to the following problem with delay terms:(9)mxutt−pxux+2δxutxx+ηθx+μutx,t−τ0=0x∈0,L,t>0,θt+kqx+ηutx=0x∈0,L,t>0,τqt+βq+kθx=0x∈0,L,t>0,u0,t=uL,t=0;θ0,t=θL,t=0,t≥0,ux,0=u0x,utx,0=u1x;θx,0=θ0x;qx,0=q0x,x∈0,L.The authors prove that the system (9) is well posed and exponential decay under a small condition on time delay (see [2]). Now in the presence of source term, the system (9) becomes the system studied in this work with a logarithmic source term; this type of problems is encountered in many branches of physics such as nuclear physics, optics, and geophysics. It is well known, from the quantum field theory, that such kind of logarithmic nonlinearity appears naturally in inflation cosmology and in supersymmetric field theories (see [11–13]).This work is organized as follows: In “Statement of Problem,” we talk briefly about the local existence of the systems (1), (2), and (3), and we define some space and theorem used. In “Blow-up of Solution,” the blow-up result is proved.


## 2. Statement of Problem

Let us introduce the function(10)zx,ρ,t=utx,t−ρτ0,x∈0,L,ρ∈0,1,t>0.Thus, we have(11)τ0ztx,ρ,t+zρx,ρ,t=0,x∈0,L,ρ∈0,1,t>0.Then, problems (1)–(3) are equivalent to(12)mxutt−pxux+2δxutxx+ηθx+μzx,1,t=uup−2lnuγ,x∈0,L,t>0,τ0ztx,ρ,t+zρx,ρ,t=0x∈0,L,ρ∈0,1,t>0,θt+kqx+ηutx=0x∈0,L,t>0,τqt+βq+kθx=0x∈0,L,t>0,(13)u0,t=uL,t=0;θ0,t=θL,t=0,t≥0,ux,0=u0x,utx,0=u1x;θx,0=θ0x,x∈0,Lqx,0=q0x,x∈0,Lzx,0,t=utx,t,x∈0,L,t>0,zx,ρ,0=f0x,−ρτ0,x∈0,L,ρ∈0,1.We first state a local existence theorem that can be established by combining the arguments of related works10,6.Letv=ut and denote by(14)Φ=u,v,θ,q,zT,Φ0=Φ0=u0,u1,θ0,q0,f0T.The state space ofΦ is the Hilbert space(15)=H010,L×L20,L×L∗20,L×L20,1×0,L.Theorem 1.
Assume that(16)2<p≤2nn−2,ifn≥3.
Then, for everyΦ0∈, there exists a unique local solution in the class Φ∈C0,T,.


## 3. Blow-up of Solution

In this section, we prove that the solutions for the problems (12)–(13) blow up in a finite time when the initial energy is negative. We use the improved method of Salim and Messaoudi [6]. We define the energy associated with problems (12)–(13) by(17)Et=12mx∞utt22+12px∞ux22+τ2q22+12θ22+τ0μ2∫01zx,ρ,t2dρ+γp2upp−1p∫0Luplnuγdx.Lemma 2.
Suppose that(18)2<p≤2nn−2,n≥3.
Then, there exists a positive constantC>0 depending on 0.L only, such that(19)∫0Luplnuγdxsp≤C∫0Luplnuγdx+ux22,for any u∈H010,L and 2≤s≤p, provided that ∫0Luplnuγdx≥0.Proof.
If∫0Luplnuγdx>1, then(20)∫0Luplnuγdxsp≤∫0Luplnuγdx.
If∫0Luplnuγdx≤1, then we set(21)Γ1=x∈0,Lu>1,and, for anyβ≤2, we have(22)∫0Luplnuγdxsp≤∫0Luplnuγdxβp≤∫Γ1uplnuγdxβp≤∫Γ1up+1dxβp≤∫0Lup+1dxβp=up+1βp+1p.
We chooseβ=2p/p+1<2 to get(23)∫0Luplnuγdxsp≤up+12≤Cux22.
Combining (20) and (23), the result was obtained.Lemma 3.
There exists a positive constantC>0 depending on 0,L only, such that(24)upp≤C∫0Luplnuγdx+ux22,for any u∈H010,L, provided that ∫0Luplnuγdx≥0.Proof.
We set(25)Γ+=x∈0,Lu>eandΓ−=x∈0,Lu≤e,
thus(26)upp=∫Γ+updx+∫Γ−updx≤∫Γ+uplnuγdx+∫Γ−epuepdx≤∫Γ+uplnuγdx+ep∫Γ−ue2dx≤∫0Luplnuγdx+ep−2∫0Lu2dx≤C∫0Luplnuγdx+ux22.By using the inequalitiesu22≤Cup2≤Cupp2/p, we have the following corollary.Corollary 4.
There exists a positive constantC>0 depending on 0,L only, such that(27)u22≤C∫0Luplnuγdx2p+ux24p.provided that ∫0Luplnuγdx≥0.Lemma 5.
There exists a positive constantC>0 depending on 0,L only, such that(28)ups≤Cupp+ux22,for any u∈H010,L and 2≤s≤p.Proof.
Ifup≥1, then(29)ups≤upp.
Ifup≤1, then ups≤up2. Using the Sobolev embedding theorems, we have(30)ups≤up2≤Cux22.
Now we are ready to state and prove our main result. For this purpose, we define(31)Ht=−Et=−12mx∞utt22−12px∞ux22−τ2q22−12θ22−τ0μ2∫01zx,ρ,t2dρ−γp2upp+1p∫0Luplnuγdx.Corollary 6.
Assume that (18) holds. Then(32)ups≤C1−γppx∞upp−2px∞Ht−mx∞px∞utt22−τpx∞q22−1px∞θ22−τ0μpx∞∫01zx,ρ,t2dρ+2ppx∞∫0Luplnuγdx,for any u∈H010,Ln and 2≤s≤p.Theorem 7.
Assume that (18) holds. Assume further that(33)E0=12mx∞u1t22+12px∞∇u022+τ2q022+12θ022+τ0μ2∫0L∫01f0x,−ρτ02dρdx+γp2u0pp−1p∫0Lu0plnu0γdx<0.
Then, the solution of (12) blows up in finite time.Proof.
we have(34)Et≤E0<0,and(35)H′t=−E′t=2δx∞uxtt22+βq22+μ∫0Lzx,1,t2dx.
Hence(36)H′t≥C0δx∞uxtt22+μ∫0Lz2x,1,tdx≥0;∀t∈0,T.
Consequently, we get(37)0<H0≤Ht≤∫0Luplnuγdx;∀t∈0,T,by virtue of (17) and (31). We then introduce(38)Lt=H1−αt+ε∫0Lmxuttut+4δxux2dx+ε∫0Lnτkuqdx,where ε>0 to be specified later and(39)2p−2p2<α<p−22p<1.
A direct differentiation ofLt gives(40)L′t=1−αH−αtH′t+ε∫0Lmxut2dx+εητk∫0Lquttdx−ε∫0Lpxux2dx+2εη∫0Lθuxdx−ε∫0Lμzx,1,tudx+ε∫0Luplnuγ−εηβk∫0Lqudx,
using the inequality of young(41)2εη∫0Lθuxdx≥−εηθ22−εηux22,(42)−ε∫0Lpxux2dx≥−εpx∞ux22,(43)ε∫0Lmxut2dx≥εmx∞ut22,(44)εητk∫0Lquttdx≥−εητ2kut22−εητ2kq22,and(45)−ε∫0Lμzx,1,tudx≥−εμξ12∫0Lzx,1,t2dx+12ξ1u22,∀ξ1>0,(46)−εηβk∫0Lqudx≥εηβkξ22q22+12ξ2u22,∀ξ2>0.
Substituting (41), (42), (43), (44), (45), and (46) in (40), we get(47)L′t≥1−αH−αtH′t+εmx∞−ητ2kut22−εpx∞+ηux22−εηθ22−ετη+βηξ22kq22+ε∫0Luplnuγdx−εμξ12∫0Lzx,1,t2dx−εu22μ2ξ1+ηβ2ξ2k.
We obtain from (35) and (47) the following:(48)L′t≥1−αH−αt−εkξ1+ηξ22kH′t+εmx∞−ητ2kut22−εpx∞+ηux22−εηθ22−ετη2kq22+ε∫0Luplnuγdx−εμk2ξ1k+ηβ2ξ2ku22.
We also setξ1=ξ2=H−αt; hence, (48) gives(49)L′t≥1−α−εCH−αtH′t+εmx∞−ητ2kut22−εpx∞+ηux22−εηθ22−ετη2kq22+ε∫0Luplnuγdx−εM2kHαtu22,where C and M are strictly positive constants depending only on k,η,β,μ.
For0<a<1, we have(50)L′t≥1−α−εCH−αtH′t+εmx∞1+p21−a−ητ2kut22+εpx∞p21−a−1+ηux22+ε−η+pε1−a2θ22+εa∫0Luplnuγdx+ε−τη2k+pτ1−a2q22+γε1−a2upp+ετ0p1−a2μ∫0L∫01zx,ρ,t2dρdx−εM2kHαtu22+pε1−aHt.
Using (27), (37) and Young’s inequality, we find(51)Hαtu22≤∫0Luplnuγdxαu22≤C∫0Luplnuγdxα+2p+∫0Luplnuγdxαux24p≤C∫0Luplnuγdxαp+2p+ux22+∫0Luplnuγdxαpp−2.
Exploiting (39), we have(52)2<αp+2≤pand2<αp2p−2≤p.
Thus, lemma 1 yields(53)Hαtu22≤C∫0Luplnuγdx+ux22.
Combining (50) and (53), we obtain(54)L′t≥1−α−εCH−αtH′t+εmx∞1+p21−a−ητ2kut22+εpx∞p21−a−1+η−CM2kux22+ε−η+pε1−a2θ22+εa−CM2k∫0Luplnuγdx+ε−τη2k+pτ1−a2q22+γε1−a2upp+ετ0p1−a2μ∫0L∫01zx,ρ,t2dρdx+pε1−aHt.
At this point, we choosea>0 so small that(55)−η+pε1−a2>0,p21−a−1>0,τ0p1−a2>0,
andk so large that(56)px∞p21−a−1+η−CM2k>0,a−CM2k>0,mx∞1+p21−a−ητ2k>0,−τη2k+pτ1−a2>0.
OnceC and a are fixed, we pick ε so small so that(57)1−α−εC>0.
Hence, (54) becomes(58)L′t≥1−α−εCH−αtH′t+εA1ut22+εA2ux22+εA3θ22+εA4q22+εa−CM2k∫0Luplnuγdx+ετ0p1−a2μ∫0L∫01zx,ρ,t2dρdx+γε1−a2upp+pε1−aHt,where A1−A4 are strictly positive constants depending only on p,τ,η,k,a.
Thus, for someA0>0, estimate (58) becomes(59)L′t≥A0Ht+ut22+ux22+uppq22+θ22+∫0Luplnuγdx+∫0L∫01zx,ρ,t2dρdx,and(60)Lt≥L0>0,∀t≥0.
Next, using Hôlder’s inequality and the embeddingu2≤Cup, we have(61)∫0Lmxuutdx≤mx∞u2ut2≤Cu2ut2,and exploiting Young’s inequality, we obtain(62)∫0Lmxuutdx11−α≤Cupr1−α+ut2r′1−α,For1r+1r′=1.
To be able to use Lemma5, we take r′=21−α which gives r/1−α=2/1−2α≤p.
Therefore, fors=2/1−2α, estimate (62) yields(63)∫0Lmxuutdx11−α≤Cups+ut22.
Hence, Lemma5 gives(64)∫0Lmxuutdx11−α≤C1upp+ut22+ux22,∀C1>0,
and with the same way, we get(65)ε∫0Lnτkuqdx11−α≤C2upp+q22,∀C2>0,(66)ε∫0L4δxux2dx11−α≤C3ux22,∀C3>0.
From (64), (65), and (66) we obtain(67)L11−αt≤CHt+upp+q22+ux22+ut22;∀t≥0,∀C>0.
Combining (67) and (59), we arrive at(68)L′t≥a0L11−αt,∀t≥0,where a0 is a positive constant depending only on A0 and C.
A simple integration of (68) over 0,t yields(69)Lα1−αt≥1L−α/1−α0−αa0t/1−α.
Therefore,Lt blows up in time(70)T∗≤1−ααa0Lα/1−α0.
The proof is completed.


## 4. Conclusion

In this work, we are interested with a problem of a logarithmic nonuniform flexible structure with time delay, where the heat flux is given by Cattaneo’s law. We show that the energy of any weak solution blows up infinite time if the initial energy is negative. The delay effect that is similar to memory processes is important in the research of applied mathematics such as physics, noninstant transmission phenomena, and biological motivation. In the future work, we will try to study the local existence for this problem with respect to some proposal conditions.



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*Source: 5555930-2021-02-11.xml*