# Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source

Authors: Hazal Yüksekkaya; Erhan Pișkin; Salah Mahmoud Boulaaras; Bahri Belkacem Cherif; Sulima Ahmed Zubair
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods.


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## Body

## 1. Introduction

In this article, we consider a plate equation with frictional damping, delay, and logarithmic terms as follows:(1)utt+Δ2u+αutt+βutx,t−τ=ulnuγforx,t∈Ω×0,∞,ux,t=∂ux,t∂υ=0forx,t∈∂Ω×0,∞,ux,0=u0x,utx,0=u1xforx∈Ω,utx,t=j0x,tforx,t∈Ω×−τ,0,where Ω⊂RN, N≥1, is a bounded domain with smooth boundary ∂Ω. τ>0 denotes time delay, and α, β, and γ are real numbers that will be specified later. Generally, logarithmic nonlinearity seems to be in supersymmetric field theories and in cosmological inflation. From quantum field theory, that kind of (uup−2lnuk) logarithmic source term seems to be in nuclear physics, inflation cosmology, geophysics, and optics (see [1, 2]). Time delays often appear in various problems, such as thermal, economic, biological, chemical, and physical phenomena. Recently, partial differential equations have become an active area with time delay (see [3, 4]). In 1986, Datko et al. [5] indicated that, in boundary control, a small delay effect is a source of instability. Generally, a small delay can destabilize a system which is uniformly stable [6]. To stabilize hyperbolic systems with time delay, some control terms will be needed (see [7–9] and references therein).For the literature review, firstly, we begin with the studies of Bialynicki-Birula and Mycielski [10, 11]. The authors investigated the equation with the logarithmic term as follows:
(2)utt−uxx+u−εulnu2=0,where the authors proved that, in any number of dimensions, wave equations including the logarithmic term have localized, stable, soliton-like solutions.In 1980, Cazenave and Haraux [12] studied the equation as follows:
(3)utt−Δu=ulnuk,where the authors in [12] proved the existence and uniqueness of the solutions for equation (3). Gorka [2] obtained the global existence results of solutions for one-dimensional equation (3). Bartkowski and Gŏrka [1] considered the weak solutions and obtained the existence results.In [13], Hiramatsu et al. studied the equation as follows:
(4)utt−Δu+u+ut+u2u=ulnu.In [14], Han established the global existence of solutions for equation (4).In [15], Al-Gharabli and Messaoudi were concerned with the plate equation with the logarithmic term as follows:
(5)utt+Δ2u+u+hut=kulnu.They established the existence results by the Galerkin method and obtained the explicit and decay of solutions utilizing the multiplier method for equation (5).In [16], Liu introduced the plate equation with the logarithmic term as follows:
(6)utt+Δ2u+utm−2ut=up−2uloguk.The author proved the local existence by the contraction mapping principle. Also, he studied the global existence and decay results. Moreover, under suitable conditions, the author proved the blow-up results withE0<0.In [17], Messaoudi studied the equation as follows:
(7)utt+Δ2u+utm−2ut=up−2u,and obtained the existence results and obtained that, if m≥p, the solution is global and blows up in finite time if m<p. Later, Chen and Zhou [18] extended this result. In the presence of the strong damping term (−Δut), Pișkin and Polat [19] proved the global existence and decay of solutions for equation (7). For more results about plate problems, see [20–22].In [7], Nicaise and Pignotti studied the equation as follows:
(8)utt−Δu+a0utx,t+autx,t−τ=0,where a0,a>0. They proved that, under the condition 0≤a≤a0, the system is exponentially stable. The authors obtained a sequence of delays that shows the solution is unstable in the case a≥a0. In the absence of delay, some other authors [23, 24] looked into exponential stability for equation (8). In [9], Xu et al., by using the spectral analysis approach, established the same result similar to [7] for the one space dimension.In [25], Nicaise et al. studied the wave equation in one space dimension in the presence of time-varying delay. In this article, the authors showed the exponential stability results with the condition
(9)a≤1−da0,where d is a constant and
(10)τ′t≤d<1,∀t>0.In [26], Kafini and Messaoudi studied wave equations with delay and logarithmic terms as follows:
(11)utt−Δu+μ1utx,t+μ2utx,t−τ=up−2uloguk.The authors proved the local existence and blow-up results for equation (11).In [27], Park considered the equation with delay and logarithmic terms as follows:
(12)utt−Δu+αutt+βutx,t−τ=ulnuγ.The author showed the local and global existence results for equation (12). Also, the author investigated the decay and nonexistence results for equation (12). In recent years, some other authors investigate hyperbolic-type equations with delay terms (see [28–33]).In this work, we studied the local existence, global existence, nonexistence, and stability results of plate equation (1) with delay and logarithmic terms, motivated by the above works. There is no research, to our best knowledge, related to plate equation (1) with the delay βutx,t−τ term and logarithmic (ulnuγ) source term; hence, our work is the generalization of the above studies.This work consists of five sections in addition to the introduction. Firstly, in Section2, we recall some assumptions and lemmas. Then, in Section 3, we obtain the local and global existence of solutions. Moreover, in Section 4, we establish the nonexistence results. Finally, in Section 5, we get the stability of solutions.


## 2. Preliminaries

In this part, we show the norm ofX by ·X for a Banach space X. We give the scalar product in L2Ω by ·,·. We show ·2 by ·, for brevity. Let B1 be the constant of the embedding inequality
(13)u2≤B1Δu2foru∈H02Ω.We have the following assumptions related to problem (1):(H1). The weights of delay and dissipation satisfy
(14)0<β<α.(H2). The constant γ in (1) satisfies
(15)0<γ<πe2N+1/N.To get the main result, we have the lemmas as follows.Lemma 1 (see [34, 35]) (Logarithmic Sobolev inequality).
For anyu∈H01Ω,
(16)∫Ωu2lnudx≤12u2lnu2+k22π∇u2−N21+lnku2,where k is a positive real number.Corollary 2.
For anyu∈H02Ω,
(17)∫Ωu2lnudx≤12u2lnu2+k22πΔu22−N21+lnku2,where k is a positive real number.Remark 3.
Assume that inequality (17) holds for all k>0, and we choose the constant k that satisfies
(18)ρ=maxe−N+1/N,μ1/Nπγ<k<πγ,where μ is any real number with
(19)0<μ<1.Lemma 4 (see [12]) (Logarithmic Gronwall inequality).
Suppose thatc>0 and l∈L10,T;R+. If a function f:0,T⟶1,∞ satisfies
(20)ft≤c1+∫0tlsfslnfsds,0≤t≤T,then
(21)ft≤cec∫0tlsds,0≤t≤T.
We define(22)Jv=12Δv2−12∫Ωv2lnvγdx+γ4v2,(23)Iv=Δv2−∫Ωv2lnvγdx,for v∈H02Ω; then,
(24)Jv=12Iv+γ4v2.
Suppose that(25)d=infv∈H02Ω\0supλ≥0Jλv.
Then, it satisfies (see, e.g., [36–38])
(26)0<d=infv∈NJv,where N is the well-known Nehari manifold, denoted by
(27)N=v∈H02Ω\0∣Iv=0.Lemma 5.
I and J are the functions that satisfy
(28)Iλv=λ∂Jλvλv>0,0<λ<λ∗,=0,λ=λ∗,<0,λ>λ∗,for any v∈H02Ω with v≠0, where
(29)λ∗=expΔv2−∫Ωv2lnvγdxγv2.Proof.
We obtain, forλ≥0,
(30)λ∂∂λJλv=λλΔv2−λ∫Ωv2lnvγdx+γλ2v2−λ∫Ωv2lnλγdx−γλ2∫Ωv2dx=λ2Δv2−∫Ωv2lnvγdx−γlnλ∫Ωv2dx=Iλv,and therefore, we obtain the desired result.☐Remark 6.
Jλv has the absolute maximum value at λ∗, such that
(31)supλ≥0Jλv=Jλ∗v=exp2Δv2−2∫Ωv2lnvγdxγv2γ4v2,for v∈H02Ω.Lemma 7.
The potential depthd in (25) satisfies
(32)d≥γ4eNπγN/2=E1.Proof.
By Corollary2, (13), and (18), we have
(33)Iv≥1−k2γ2πΔv2+Nγ21+lnkv2−γ2v2lnv2>Nγ21+lnkv2−γ2v2lnv2.
Taking the limitk⟶π/γ, we obtain
(34)Iv≥Nγ21+lnπγ−γ2lnv2v2.
Taking into consideration this and (28), we get
(35)0=Iλ∗v≥Nγ21+lnπγ−γ2lnλ∗v2λ∗v2,and therefore,
(36)λ∗v2≥eNπγN/2.
Hence, we have by (24) and (31)
(37)supλ≥0Jλv=Jλ∗v=12Iλ∗v+γ4λ∗v2=γ4λ∗v2≥γ4eNπγN/2.
From the definition ofd given in (25), we obtain the result.☐


## 3. Existence

In this part, we have studied the local existence, global existence, nonexistence, and stability results of plate equation (1) with delay and logarithmic terms, motivated by the above works. There is no research, to our best knowledge, related to plate equation (1) with the delay (βutx,t−τ) term and logarithmic (ulnuγ) source term; hence, our work is the generalization of the above studies. Firstly, we introduce the new function
(38)yx,η,t=utx,t−ητforx,η,t∈Ω×0,1×0,∞.Hence, problem (1) takes the form
(39)utt+Δ2u+αutx,t+βyx,1,t=ulnuγforx,t∈Ω×0,∞,τytx,η,t+yηx,η,t=0forx,η,t∈Ω×0,1×0,∞,ux,t=∂ux,t∂υ=0forx,t∈∂Ω×0,∞,ux,0=u0x,utx,0=u1xforx∈Ω,yx,η,0=j0x,−ητ=y0x,ηforx,η∈Ω×0,1.Definition 8.
Assume thatT>0. u,y is a local solution of problem (39) if it satisfies
(40)u∈C0,T;H02Ω∩C10,T;L2Ω∩C20,T;H−2Ω,utt,v+Δu,Δv+αutt,v+βy1,t,v=ulnuγ,vforanyv∈H02Ω,τ∫01ytη,t,φηdη+∫01yηη,t,φηdη=0foranyφ∈L2Ω×0,1,u0=u0inH02Ω,ut0=u1inL2Ω,y0=y0inL2Ω×0,1.


### 3.1. Local Existence

In this part, we establish the local existence results similar to [8, 39].Theorem 9.
Suppose that (H1) and (H2) are satisfied. Then, for the initial datau0∈H02Ω, u1∈L2Ω, and y0∈L2Ω×0,1, there exists a local solution u,y for problem (39).Proof.
Letvii∈N be the orthogonal basis of H02Ω that is orthonormal in L2Ω. Define φix,0=vix, and we extend φix,0 by φix,η over L2Ω×0,1. We denote Vn=spanv1,v2,⋯,vn and Wn=spanφ1,φ2,⋯,φn for n≥1. We consider the Faedo-Galerkin approximation solution un,yn∈Vn×Wn of the form
(41)un=∑ni=1hintvix,ynx,η,t=∑ni=1gintφix,η,n=1,2,⋯,solving the approximate system
(42)uttn,v+Δun,Δv+αutnt,v+βyn1,t,v=∫Ωunlnunγvdxforv∈Vn,(43)τ∫01ytnη,t,φηdη+∫01yηnη,t,φηdη=0forφ∈Wn,(44)un0=u0n,utn0=u1n,yn0=y0n,where
(45)u0n⟶u0inH02Ω,u1n⟶u1inL2Ω,y0n⟶y0inL2Ω×0,1.
Since problem (42)–(44) is a normal system of ordinary differential equations, there exists a solution un,yn on the interval 0,tn, tn∈0,T. The extension of that solution to the 0,T is a consequence of the estimate below.
By replacingv by utnt in (42) and by using the relation
(46)∫Ωunlnunγutndx=ddt12∫Ωun2lnunγdx−γ4un2,we have
(47)ddt12utn2+12Δun2+γ4un2−12∫Ωun2lnunγdx=−αutnt2−βyn1,t,utnt.
By replacingφ by ωynη,t in (43), we see that
(48)ωτ2ddt∫Ω∫01ynx,η,t2dηdx=−ω2yn1,t2+ω2yn0,t2.
Summing (47) and (48), we obtain
(49)ddtEnt=−αutn2−βyn1,t,utnt−ω2yn1,t2+ω2yn0,t2,where
(50)Ent=12utn2+12Δun2+γ4un2−12∫Ωun2lnunγdx+ωτ2ynL2Ω×0,12,where
(51)β<ω<2α−β.
Utilizing Young’s inequality and the fact thatynx,0,t=utnx,t, we obtain
(52)ddtEnt≤−α−β2−ω2utn2−ω2−β2yn1,t2≤0,(53)Ent+C1∫0tutns2ds+C2∫0tyn1,s2ds≤En0,where
(54)C1=α−β2−ω2>0,C2=ω2−β2>0.
Taking into consideration this and Corollary2, we have
(55)utn2+1−γk22πΔun2+γ21+N1+lnkun2+2C1∫0tutns2ds+2C2∫0tyn1,s2ds+ωτynL2Ω×0,12≤2En0+γ2un2lnun2.
By using (18), we obtain
(56)1−γk22π>0,γ21+N1+lnk>0,and therefore,
(57)utn2+Δun2+un2+∫0tutns2ds+∫0tyn1,s2ds+ynL2Ω×0,12≤c11+un2lnun2,where the sequel cj, j=1,2,⋯, shows a positive constant. Also, we know that
(58)unx,t=unx,0+∫0tutnx,sds.
Utilizing Cauchy-Schwarz’s inequality and (57), we obtain
(59)unt2=2un02+2T∫0tutns2ds≤2un02+2T∫0tc11+uns2lnuns2ds≤c21+∫0tuns2lnuns2ds.
From Lemma4, we arrive at
(60)unt2≤c3ec4T.
fs=slns is the function which is continuous on 0,∞, lims⟶0+fs=0, lims⟶+∞fs=+∞, and f decreases on 0,e−1 and increases on e−1,+∞; hence, we get by (57) and (60)
(61)utn2+Δun2+un2+∫0tutns2ds+∫0tyn1,s2ds+ynL2Ω×0,12≤c5.
Hence, there exists a subsequence ofun,yn, which we still denote un,yn, such that
(62)un⟶uweaklystarinL∞0,T;H02Ω,utn⟶utweaklystarinL∞0,T;L2Ω,yn⟶yweaklystarinL∞0,T;L2Ω×0,1,yn1⟶y1weaklyinL20,T;L2Ω.
Utilizing the Aubin-Lions compactness theorem, we conclude that(63)un⟶ustronglyinL20,T;L2Ω,un⟶ua.e.inΩ×0,T.
The functions⟶slnsγ is continuous on R; hence,
(64)unlnunγ⟶ulnuγa.e.inΩ×0,T.
Let(65)Ω1=x∈Ω∣un<1,Ω2=x∈Ω∣un≥1.
Thus, we obtain(66)∫Ωunlnunγ2dx=γ2∫Ω1unlnun2dx+∫Ω2unlnun2dx≤γ2e−2Ω1+e−22q−22∫Ω2unqdxforanyq>2,where we used
(67)slns≤1efor0<s<1,s−κlns≤1eκfors≥1andκ>0.
By (57) and (66), we conclude that
(68)∫Ωunlnunγ2dx≤γ2e−2Ω1+e−22q−22B2qΔunq≤c6,where B2 is the Sobolev imbedding constant of
(69)H02Ω⊂LqΩforq>2,ifN=1,2,3,4;2<q<2NN−4,ifN≥5.
Therefore, we get from (68)
(70)unlnunγwhichisuniformlyboundedinL∞0,T;L2Ω.
From the Lebesgue bounded convergence theorem, (64), and (70), we arrive at
(71)unlnunγ⟶ulnuγstronglyinL20,T;L2Ω.
We pass the limitm⟶∞ in (42) and (43). The remainder of the proof is standard and similar to [39, 40].☐


### 3.2. Global Existence

In this part, we obtain the global existence results for problem (39). For this goal, we define the energy functional of problem (39):
(72)Et=12ut2+12Δu2+γ4u2−12∫Ωu2lnuγdx+ωτ2yL2Ω×0,12,where ω is the positive constant given in (51). We see that
(73)Et=12ut2+Jut+ωτ2yL2Ω×0,12=12ut2+12Iut+γ4u2+ωτ2yL2Ω×0,12.By the same arguments similar to (52), we infer that
(74)ddtEt≤−C1ut2−C2y1,t2≤0,where C1 and C2, given in (54), are positive constants.Lemma 10.
Suppose that (H1) and (H2) are satisfied. IfE0<d and Iu0>0, then the solution u of problem (1) satisfies
(75)Iut>0fort∈0,T,where T is the maximal existence time of the solutions.Proof.
We know thatIu0>0 and u is continuous on 0,T; hence, we have
(76)Iut>0forsomeinterval0,t1∈0,T.
Lett0 be the maximum of t1 satisfying (76). Assume that t0<T; then, Iut0=0, that is,
(77)ut0∈N.
Therefore, we obtain by (26)
(78)Jut0≥infv∈NJv=d.
We see that this is in contradiction to the relation as follows:(79)Jut0≤Et0≤E0<d.
By (74) and Lemma 10, we see that Et is a nonincreasing function.☐Theorem 11.
The solutionu is global, under the conditions of Lemma 10.Proof.
It suffices to show thatut2+Δu2 is bounded independent of t. By Lemma 10, (73), and (74), we get
(80)ut2≤ut2+Iut≤2Et≤2E0<2d.
In a similar way, we get(81)u2<u2+2γIut=4γJut≤4γEt≤4γE0<4dγ.
By Corollary2 and (23), we conclude that
(82)Δu2=Iut+γ∫Ωu2lnudx≤2Et+γ2u2lnu2+k2γ2πΔu2−Nγ21+lnku2.
By taking the limitk⟶ρ+ in this inequality and from (81), we obtain
(83)1−ρ2γ2πΔu2≤2Et+γ2lnu2−N1+lnρu2<2d+γ2ln4dγ−N1+lnρu2=2d+γ2ln4dγe−Nρ−Nu2.
By Lemma7 and (18), we get
(84)ln4dγe−Nρ−N≥lnπγN/2ρ−N=lnπγρ−1Nln1=0.
Therefore, we see by (81) and (83) that
(85)1−ρ2γ2πΔu2≤2d+2dln4dγe−Nρ−N.
Hence, we conclude that(86)Δu2<2d1−ρ2γ2π−11+ln4dγe−Nρ−N.
Therefore, we complete the proof by (80) and (86).☐


## 3.1. Local Existence

In this part, we establish the local existence results similar to [8, 39].Theorem 9.
Suppose that (H1) and (H2) are satisfied. Then, for the initial datau0∈H02Ω, u1∈L2Ω, and y0∈L2Ω×0,1, there exists a local solution u,y for problem (39).Proof.
Letvii∈N be the orthogonal basis of H02Ω that is orthonormal in L2Ω. Define φix,0=vix, and we extend φix,0 by φix,η over L2Ω×0,1. We denote Vn=spanv1,v2,⋯,vn and Wn=spanφ1,φ2,⋯,φn for n≥1. We consider the Faedo-Galerkin approximation solution un,yn∈Vn×Wn of the form
(41)un=∑ni=1hintvix,ynx,η,t=∑ni=1gintφix,η,n=1,2,⋯,solving the approximate system
(42)uttn,v+Δun,Δv+αutnt,v+βyn1,t,v=∫Ωunlnunγvdxforv∈Vn,(43)τ∫01ytnη,t,φηdη+∫01yηnη,t,φηdη=0forφ∈Wn,(44)un0=u0n,utn0=u1n,yn0=y0n,where
(45)u0n⟶u0inH02Ω,u1n⟶u1inL2Ω,y0n⟶y0inL2Ω×0,1.
Since problem (42)–(44) is a normal system of ordinary differential equations, there exists a solution un,yn on the interval 0,tn, tn∈0,T. The extension of that solution to the 0,T is a consequence of the estimate below.
By replacingv by utnt in (42) and by using the relation
(46)∫Ωunlnunγutndx=ddt12∫Ωun2lnunγdx−γ4un2,we have
(47)ddt12utn2+12Δun2+γ4un2−12∫Ωun2lnunγdx=−αutnt2−βyn1,t,utnt.
By replacingφ by ωynη,t in (43), we see that
(48)ωτ2ddt∫Ω∫01ynx,η,t2dηdx=−ω2yn1,t2+ω2yn0,t2.
Summing (47) and (48), we obtain
(49)ddtEnt=−αutn2−βyn1,t,utnt−ω2yn1,t2+ω2yn0,t2,where
(50)Ent=12utn2+12Δun2+γ4un2−12∫Ωun2lnunγdx+ωτ2ynL2Ω×0,12,where
(51)β<ω<2α−β.
Utilizing Young’s inequality and the fact thatynx,0,t=utnx,t, we obtain
(52)ddtEnt≤−α−β2−ω2utn2−ω2−β2yn1,t2≤0,(53)Ent+C1∫0tutns2ds+C2∫0tyn1,s2ds≤En0,where
(54)C1=α−β2−ω2>0,C2=ω2−β2>0.
Taking into consideration this and Corollary2, we have
(55)utn2+1−γk22πΔun2+γ21+N1+lnkun2+2C1∫0tutns2ds+2C2∫0tyn1,s2ds+ωτynL2Ω×0,12≤2En0+γ2un2lnun2.
By using (18), we obtain
(56)1−γk22π>0,γ21+N1+lnk>0,and therefore,
(57)utn2+Δun2+un2+∫0tutns2ds+∫0tyn1,s2ds+ynL2Ω×0,12≤c11+un2lnun2,where the sequel cj, j=1,2,⋯, shows a positive constant. Also, we know that
(58)unx,t=unx,0+∫0tutnx,sds.
Utilizing Cauchy-Schwarz’s inequality and (57), we obtain
(59)unt2=2un02+2T∫0tutns2ds≤2un02+2T∫0tc11+uns2lnuns2ds≤c21+∫0tuns2lnuns2ds.
From Lemma4, we arrive at
(60)unt2≤c3ec4T.
fs=slns is the function which is continuous on 0,∞, lims⟶0+fs=0, lims⟶+∞fs=+∞, and f decreases on 0,e−1 and increases on e−1,+∞; hence, we get by (57) and (60)
(61)utn2+Δun2+un2+∫0tutns2ds+∫0tyn1,s2ds+ynL2Ω×0,12≤c5.
Hence, there exists a subsequence ofun,yn, which we still denote un,yn, such that
(62)un⟶uweaklystarinL∞0,T;H02Ω,utn⟶utweaklystarinL∞0,T;L2Ω,yn⟶yweaklystarinL∞0,T;L2Ω×0,1,yn1⟶y1weaklyinL20,T;L2Ω.
Utilizing the Aubin-Lions compactness theorem, we conclude that(63)un⟶ustronglyinL20,T;L2Ω,un⟶ua.e.inΩ×0,T.
The functions⟶slnsγ is continuous on R; hence,
(64)unlnunγ⟶ulnuγa.e.inΩ×0,T.
Let(65)Ω1=x∈Ω∣un<1,Ω2=x∈Ω∣un≥1.
Thus, we obtain(66)∫Ωunlnunγ2dx=γ2∫Ω1unlnun2dx+∫Ω2unlnun2dx≤γ2e−2Ω1+e−22q−22∫Ω2unqdxforanyq>2,where we used
(67)slns≤1efor0<s<1,s−κlns≤1eκfors≥1andκ>0.
By (57) and (66), we conclude that
(68)∫Ωunlnunγ2dx≤γ2e−2Ω1+e−22q−22B2qΔunq≤c6,where B2 is the Sobolev imbedding constant of
(69)H02Ω⊂LqΩforq>2,ifN=1,2,3,4;2<q<2NN−4,ifN≥5.
Therefore, we get from (68)
(70)unlnunγwhichisuniformlyboundedinL∞0,T;L2Ω.
From the Lebesgue bounded convergence theorem, (64), and (70), we arrive at
(71)unlnunγ⟶ulnuγstronglyinL20,T;L2Ω.
We pass the limitm⟶∞ in (42) and (43). The remainder of the proof is standard and similar to [39, 40].☐


## 3.2. Global Existence

In this part, we obtain the global existence results for problem (39). For this goal, we define the energy functional of problem (39):
(72)Et=12ut2+12Δu2+γ4u2−12∫Ωu2lnuγdx+ωτ2yL2Ω×0,12,where ω is the positive constant given in (51). We see that
(73)Et=12ut2+Jut+ωτ2yL2Ω×0,12=12ut2+12Iut+γ4u2+ωτ2yL2Ω×0,12.By the same arguments similar to (52), we infer that
(74)ddtEt≤−C1ut2−C2y1,t2≤0,where C1 and C2, given in (54), are positive constants.Lemma 10.
Suppose that (H1) and (H2) are satisfied. IfE0<d and Iu0>0, then the solution u of problem (1) satisfies
(75)Iut>0fort∈0,T,where T is the maximal existence time of the solutions.Proof.
We know thatIu0>0 and u is continuous on 0,T; hence, we have
(76)Iut>0forsomeinterval0,t1∈0,T.
Lett0 be the maximum of t1 satisfying (76). Assume that t0<T; then, Iut0=0, that is,
(77)ut0∈N.
Therefore, we obtain by (26)
(78)Jut0≥infv∈NJv=d.
We see that this is in contradiction to the relation as follows:(79)Jut0≤Et0≤E0<d.
By (74) and Lemma 10, we see that Et is a nonincreasing function.☐Theorem 11.
The solutionu is global, under the conditions of Lemma 10.Proof.
It suffices to show thatut2+Δu2 is bounded independent of t. By Lemma 10, (73), and (74), we get
(80)ut2≤ut2+Iut≤2Et≤2E0<2d.
In a similar way, we get(81)u2<u2+2γIut=4γJut≤4γEt≤4γE0<4dγ.
By Corollary2 and (23), we conclude that
(82)Δu2=Iut+γ∫Ωu2lnudx≤2Et+γ2u2lnu2+k2γ2πΔu2−Nγ21+lnku2.
By taking the limitk⟶ρ+ in this inequality and from (81), we obtain
(83)1−ρ2γ2πΔu2≤2Et+γ2lnu2−N1+lnρu2<2d+γ2ln4dγ−N1+lnρu2=2d+γ2ln4dγe−Nρ−Nu2.
By Lemma7 and (18), we get
(84)ln4dγe−Nρ−N≥lnπγN/2ρ−N=lnπγρ−1Nln1=0.
Therefore, we see by (81) and (83) that
(85)1−ρ2γ2πΔu2≤2d+2dln4dγe−Nρ−N.
Hence, we conclude that(86)Δu2<2d1−ρ2γ2π−11+ln4dγe−Nρ−N.
Therefore, we complete the proof by (80) and (86).☐


## 4. Nonexistence

In this part, similar to [41–43], we get the nonexistence results for problem (1). Firstly, we need the lemma as follows.Lemma 12.
Assume that (H1) and (H2) are satisfied. IfE0<E1 and Iu0<0, then the solution u of problem (1) satisfies
(87)Iut<0fort∈0,T,(88)ut2>4E1γfort∈0,T,where T is the maximal existence time of the solutions.Proof.
We know thatIu0<0 and u is continuous on 0,T; hence, we have
(89)Iut<0forsomeinterval0,t1⊂0,T.
Lett0 be the maximal time satisfying (89) and assume that t0<T; then, Iu0=0, such that
(90)ut0∈N.
Therefore, we obtain(91)d≤Jut0=12Iut0+γ4ut02≤Eut0≤E0<E1.
This is in contradiction to Lemma7. Thus, (87) is proved. By Lemma 7, (31), and (87), we conclude that
(92)E1≤d≤Jλ∗ut=exp2Δu2−2∫Ωu2lnuγdxγu2γ4u2<γ4u2.
Therefore, the proof is completed.☐Theorem 13.
Suppose that (H1) and (H2) are satisfied. LetE0<ζE1, where 0<ζ<1, and Iu0<0. Then, the solution of problem (1) blows up at infinity.Proof.
Firstly, we set(93)Ft=ζE1−Et.
By (74), we obtain
(94)F′t=−E′t≥C1ut2+C2y1,t2≥0.
Utilizing (72), (88), and (94), we see that
(95)0<F0≤Ft≤ζE1+12∫Ωu2lnuγdx<γ4u2+12∫Ωu2lnuγdx.
We define(96)Gt=Ft+εu,ut+εα2u2.
By (39) and (72), we get
(97)G′t=F′t+εut2−εΔu2−εβu,y1,t+ε∫Ωu2lnuγdx=F′t+2εut2−εβu,y1,t−2εEt+εγ2u2+ωτyL2Ω×0,12.
Utilizing Young’s inequality and (94), we obtain
(98)βu,y1,t≤βδu2+14δy1,t2≤δβu2+β4δC2F′t.
By adapting this to (97) and from (88) and (93), we have
(99)G′t≥1−εβ4δC2F′t+2εut2+εγ2−εβδu2+2εFt−2εζE1+ωτyL2Ω×0,12≥1−εβ4δC2F′t+2εut2+ε1−ζγ2−βδu2+2εFt+ωτyL2Ω×0,12.
Firstly, fixδ>0 such that 1−ζγ/2−βδ>0 and then choose ε>0 small enough so that 1−εβ/4δC2>0. Then, by (94), we get
(100)G′t≥c8Ft+ut2+u2≥0.
Also, we conclude that(101)Gt≤c9Ft+ut2+u2.
Takingε>0 small enough again, we obtain
(102)G0=F0+εu0,u1+εα2u02>0.
By (100) and (102), we get
(103)Gt≥G0>0.
Utilizing (100) and (101), we see that
(104)G′t≥c10Gt,and therefore,
(105)Gt≥ec10tG0>0.
Therefore,Gt blows up at infinity. Consequently, the proof is completed.☐


## 5. Stability

In this part, we obtain the stability of global solutions. Firstly, we define the perturbed energy by(106)Ψt=Et+εΦt+εΞt,where ε>0, Φt=ut,u, and Ξt=∫Ω∫01e−τηy2x,η,tdηdx.Lemma 14.
Under the conditions of Lemma10, for C3,C4>0, we obtain
(107)C3Et≤Ψt≤C4Et.Proof.
Utilizing Lemma10 and Young’s inequality, we have
(108)Φt+Ξt≤12ut2+12u2+yL2Ω×0,12≤12ut2+2γγ4u2+12Iut+yL2Ω×0,12=12ut2+2γJut+yL2Ω×0,12≤c7Et.
Takingε>0 small enough, we complete the proof.☐Theorem 15.
Assume that (H1) and (H2) are satisfied. Suppose thatE0<E1 and Iu0>0. Hence, for C0,C5>0, we obtain
(109)0<Et≤C0e−C5tfort≥0.Proof.
From (39) and Young’s inequality, we get
(110)Φ′t=ut2−Δu2−αutt,ut−βy1,t,ut+∫Ωu2lnuγdx≤ut2−12Δu2+α2B1utt2+β2B1y1,t2+∫Ωu2lnuγdx.
By using the second equation of (39) and the integration by parts, we obtain
(111)Ξ′t=−2τ∫Ω∫01e−τηyx,η,tyηx,η,tdηdx=−1τ∫Ω∫01e−τη∂∂ηy2x,η,tdηdx=−e−ττy1,t2+1τy0,t2−∫Ω∫01e−τηy2x,η,tdηdx≤1τut2−e−τ∫Ω∫01y2x,η,tdηdx.
Summing these and (74), we obtain
(112)Ψ′t≤−C1−ε−εα2B1−ετut2−ε2Δu2−C2−εβ2B1y1,t2+ε∫Ωu2lnuγdx−εe−τyL2Ω×0,12.
Adding and subtractingξEt with 0<ξ<2ε, we get
(113)Ψ′t≤−ξEt−C1−ε−εα2B1−ετ−ξ2ut2−ε2−ξ2−ξγB14Δu2−C2−εβ2B1y1,t2+ε−ξ2∫Ωu2lnuγdx−εe−τ−ξωτ2yL2Ω×0,12.
Utilizing the logarithmic Sobolev inequality, we have(114)Ψ′t≤−ξEt−C1−ε−εα2B1−ετ−ξ2ut2−ε12−γk22π−ξ21−γk22π−ξγB14Δu2+γ2ε−ξ2lnu2−N1+lnku2−C2−εβ2B1y1,t2−εe−τ−ξωτ2yL2Ω×0,12.
Now, chooseε>0 small enough, such that
(115)C1−ε−εα2B1−ετ>0,C2−εβ2B1>0.
By takingξ>0 sufficiently small and noting that 1/2−γk2/2π>0 (see (18)), we infer that
(116)Ψ′t≤−ξEt+γ2ε−ξ2lnu2−N1+lnku2,where 0<E0<E1; therefore, there exists 0<μ<1, that is, E0=μE1. Therefore, we obtain by (81)
(117)lnu2<ln4γEt≤ln4γE0=ln4μE1γ=lnμeNπγN/2.
Hence, by (18), we arrive at
(118)lnu2−N1+lnk≤lnμeNπγN/2−N1+lnk=Nlnμ1/Nπγk−1<Nln1=0.
Substituting this into (116), we arrive at
(119)Ψ′t≤−ξEt.
As a result, from Lemma14, we completed the proof.☐


## 6. Conclusions

Recently, there have been many published works related to wave equations with time delay. There were no local existence, global existence, nonexistence, and stability results of the plate equation with delay and logarithmic source terms, to the best of our knowledge. Firstly, we have obtained the local and global existence results. Then, we have obtained the nonexistence of solutions. Finally, we have proved stability results under sufficient conditions.



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*Source: 8561626-2021-06-28.xml*