# On a Nonhomogeneous Timoshenko System with Nonlocal Constraints

Authors: Said Mesloub; Faten Aldosari
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Our main concern in this paper is to prove the well posedness of a nonhomogeneous Timoshenko system with two damping terms. The system is supplemented by some initial and nonlocal boundary conditions of integral type. The uniqueness and continuous dependence of the solution on the given data follow from some established a priori bounds, and the proof of the existence of the solution is based on some density arguments.


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## Body

## 1. Introduction

Timoshenko [1] was the first who introduced a model describing the transverse vibration of a beam. More precisely, his research work concerns with the correction for shear of a differential equation for transverse vibrations of prismatic bars. This model was given by a system of two coupled hyperbolic partial differential equations complemented with some boundary conditions.
(1)ρ1utt=κux−vxx,t∈0,L×0,∞,ρ2vtt=κ∗vxx+κux−vx,t∈0,L×0,∞,ux−vx=0x=L=0,vxx=0x=L=0,where L is the length of the beam in its equilibrium configuration. The function u models the transverse displacement of the beam, and v models the rotation angle of its filament. The coefficients ρ1,ρ2,κ, and κ∗ are, respectively, the density, the polar moment of inertia of a cross section, the shear modulus, and the Young’s modulus of elasticity. In [2], the authors considered and proved some exponential decay results for a linear homogeneous Timoshenko system with a memory term of the form
(2)ρ1Utt−κ1Ux+Vx=0,ρ2Vtt−κ2Vxx+κ1Ux+V+h∗Vxxx,t=0,U0,t=UL,t=V0,t=VL,t=0,Ux,0=U0,Utx,0=U1,Vx,0=V0,Vtx,0=V1,where x,t∈0,L×0,∞. The same problem (2) was considered in [3] where the authors discussed the decay properties of the semigroup generated by a linear Timoshenko system with fading memory. In paper [4], the authors studied the exponential stability for the following Timoshenko system with two weak dampings.
(3)ρ1utt=κux−vx−ut,in0,L×0,∞,ρ2vtt=κ∗vxx−κux−vx−vt,in0,L×0,∞,u0,t=uL,t=v0,t=vL,t=0,t>0.In [5], the authors investigated the effect of both frictional and viscoelastic dampings. They considered in the domain 0,L×0,∞ the following system
(4)Utt−Ux+Vx=0,Vtt−Vxx+Ux+V+∫0tgt−saxvxx,sxds+bxhvt=0,u0,t=u1,t=v0,t=v1,t=0,t>0,and proved some exponential and polynomial decay results. For more results concerning Timoshenko systems, we refer the reader to [6–15].Motivated by the above systems, we consider a nonlocal initial boundary value problem for a nonhomogeneous Timoshenko system with memory term of type (2), complemented with boundary integral boundary conditions. The study of mixed problems with nonlocal conditions such as integral conditions goes back to the year 1963, when Cannon [16] used the potential method to investigate the existence and uniqueness of the solution of the heat equation subject to the specification of energy (integral constraint). This type of conditions arises mainly when the data cannot be measured directly on the boundary, but only their averages (weighted averages) are known. Due to their importance, physical significance (mean, total flux, total energy, etc.), and numerous applications in different fields of science and engineering, several authors extensively studied this type of problems, and we can cite, for example, [17–24]. Some recent new results on this direction were obtained (see [25, 26]). In this work, a functional analysis method based on some a priori bounds and on the density of the range of the unbounded operator corresponding to the abstract formulation of the given problem is used to prove the well posedness of the posed problem. This work can be considered as a contribution to the development of the energy inequality method used to prove the well posedness of mixed problems with nonlocal conditions such as integral boundary conditions (see, for example, [17, 18, 27–31]).


## 2. Formulation of the Problem and Function Spaces

In the bounded domainQT=0,L×0,T, we consider the initial boundary value problem for a nonhomogeneous Timoshenko system with a viscoelastic term of the form
(5)L1u,v=Fx,t,in0,L×0,T,L2u,v=Gx,t,in0,L×0,T,where
(6)L1u,v=ρ1utt−κ1ux+vx+ut,L2u,v=ρ2vtt−κ2vxx+κ1ux+v+∫0tht−svxxx,sds,ρ1,ρ2,κ1, and κ2 are positive constants, f,g,φ,ψ,F, and Gare given functions, and h:ℝ+→ℝ+ is a twice differentiable function such that
(7)κ2−∫0Thtdt=l>0,h′t<0,∀t≥0.The convolution term(8)h∗vxxx,t=∫0tht−svxxx,sds,represents the memory effect with a real valued function h of class C2.System (5) is supplemented with the initial conditions
(9)ℓ1u=ux,0=φx,ℓ2u=utx,0=ψx,ℓ1v=vx,0=fx,ℓ2v=vtx,0=gx,and the boundary integral conditions
(10)∫0Ludx=0,∫0Lxudx=0,∫0Lvdx=0,∫0Lxvdx=0.This system of coupled hyperbolic equations represents a Timoshenko model for a thick beam of lengthL,where u is the transverse displacement of the beam and v is the rotation angle of the filament of the beam. The coefficients ρ1,ρ2,κ1, and κ2 are, respectively, the density, the polar moment of inertia of a cross section, the shear modulus, and the Young modulus of elasticity. The integral conditions represent the averages (weighted averages) of the total transverse displacement of the beam and the rotation angle of the filament of the beam.Our aim is to study the well posedness of the solution of problems (5), (9), and (10). That is, on the basis of some a priori bounds and on the density of the range of the operator generated by the problem under consideration, we prove the existence, uniqueness, and continuous dependence of the solution on the given data of problems (5), (9), and (10). We now introduce some function spaces needed throughout the sequel. Let L2QT be the Hilbert space of square integrable functions on QT=0,1×0,T,T<∞, with scalar product and norm, respectively.
(11)Z,SL2QT=∫QTZSdxdt,ZL2QT2=∫QTZ2dxdt.We also use the spaceL20,1 on the interval 0,1, whose definition is analogous to the space on Q. Let B210,L be the space obtained by completion of the space C00,L of real continuous functions with compact support in the interval 0,L with respect to the inner product
(12)θ,θ∗B210,L=∫0LIxθ.Ixθ∗dx,where Ixθ=∫0xθζdζ for every fixed x∈0,L. The associated norm is θB210,L2=θ,θB210,L=∫0LIxθ2dx. We denote by CJ¯;L20,L, with J=0,T the set of all continuous functions θ.,t:J→L20,L with norm
(13)θCJ;L20,L2=sup0≤t≤Tθ.,tL20,L2<∞,and CJ¯;B210,L the set of functions θ.,t:J¯→B210,L with norm
(14)θCJ¯;B210,L2=sup0≤t≤TIxθ.,tL20,L2=sup0≤t≤Tθ.,tB210,L2<∞.To obtain a priori estimates for the solution, we write down our problems (5), (9), and (10) in its operator form: AU=H with U=u,v, AU=L1u,v,L2u,v, and H=H1,H2, where
(15)L1u,v=L1u,v,ℓ1u,ℓ2u,L2u,v=L2u,v,ℓ1v,ℓ2vH1=F,φ,ψ,H2=G,f,g.,The operatorA is an unbounded operator of domain of definition DA consisting of elements u,v∈L2J¯;L20,L2 such that ux,vx,ut,vt,utt,vtt,uxx,vxx belong to L2J¯;L20,L and verify initial and boundary conditions (9) and (10). The operator A is acting from the Banach space B into the Hilbert space E, where B is the Banach space obtained by completing DA with respect to the norm
(16)∥U∥B2=ut.,tCJ¯;B210,L2+vt.,tCJ¯;B210,L2+∥u.,t∥CJ¯;L20,L2+∥v.,t∥CJ¯;L20,L2,and E=L2QT×L20,L2×L2QT×L20,L2 is the Hilbert space consisting of vector-valued functions H=F,φ,ψ,G,f,g for which the norm
(17)∥H∥E2=∥F∥L2QT2+∥φ∥L20,L2+∥ψ∥L20,L2+∥G∥L2QT2+∥f∥L20,L2+∥g∥L20,L2.is finite. The functions U=u,v are continuous on the interval J¯ with values in L20,L and having continuous derivatives Ut=ut,vt on J¯ with values in B210,L. Hence, the mappings
(18)ℓ1:B∋U=u,v→ℓ1Ut=0=ℓ1u,ℓ1v=u,vt=0∈L20,L×L20,L,ℓ2:B∋Ut=ut,vt→ℓ2Ut=0=ℓ2u,ℓ2v=ut,vtt=0∈B210,L×B210,L,are defined and continuous on the Banach space B.


## 3. A Priori Estimate and Its Consequences

In this section, we establish an energy inequality from which we deduce the uniqueness and continuous dependence of solution of problems (5), (9), and (10) on the given data.Theorem 1.
For any functionU=u,v∈DA, the following a priori estimate holds
(19)ut.,tCJ¯;B210,L+vt.,tCJ¯;B210,L+∥u.,t∥CJ¯;L20,L2+∥v.,t∥CJ¯;L20,L2≤C∥φ∥L20,L2+∥ψ∥L20,L2+∥g∥L20,L2+∥f∥L20,L2+∥F∥L2QT2+∥G∥L2QT2,where C=DeDT with D is a positive constant independent of U≪u,v given by equation (41) below.Proof.
Define the integrodifferential operatorsM1u=−Ix2ut and M2v=−Ix2vt, where
(20)Ix2ux,t=∫0x∫0ξuη,tdηdξ,Ix2vx,t=∫0x∫0ξvη,tdηdξ,and consider the identity
(21)ρ1utt,M1uL2Qτ−κ1ux+vx,M1uL2Qτ−ut,M1uL2Qτ+ρ2vtt,M2vL2Qτ−κ2vxx,M2vL2Qτ+κ1ux+v,M2vL2Qτ+∫0tht−svxxx,sds,M2vL2Qτ=Fx,t,M1uL2Qτ+Gx,t,M2vL2Qτ,where Qτ=0,L×0,τ.The standard integration by parts of each term in (21) and conditions (9) and (10) give
(22)−ρ1utt,Ix2utL2Qτ=−ρ1∫0τ∫0LuttIx2utdxdt=−ρ1∫0τIxuttIx2ut0Ldt+ρ1∫0τ∫0LIxuttIxutdxdt=ρ12∥Ixut.,τ∥L20,L2−∥Ixψ∥L20,L2,(23)−ρ2vtt,Ix2vtL2Qτ=ρ22∥Ixvt.,τ∥L20,L2−∥Ixg∥L20,L2,(24)−ut,M1uL2Qτ=∥Ixut∥L2Qτ2,(25)κ1uxx,Ix2utL2Qτ=κ1∫0τ∫0Luxx.Ix2utdxdt=κ1∫0τIx2ut.ux0Ldt−κ1∫0τ∫0LIxut.uxdxdt=κ1∫0τ∫0Lutudxdt=κ12∥u.,τ∥L0,L22−∥φ∥L0,L22,and in the same manner, we have
(26)κ2vxx,Ix2vtL2Qτ=κ22v.,τL0,L22−fL0,L22,(27)κ1vx,Ix2utL2Qτ=κ1∫0τ∫0Lvx.Ix2utdxdt=κ1∫0τIx2ut.v0Ldt−κ1∫0τ∫0LIxut.vdxdt=−κ1∫0τ∫0LIxut.vdxdt,(28)−κ1ux,Ix2vtL2Qτ=κ1∫0τ∫0LIxvt.udxdt,(29)−κ1v,Ix2vtL2Qτ=−κ1∫0τ∫0Lv.Ix2vtdxdt=−κ1∫0τIx2vt.Ixv0Ldt+κ1∫0τ∫0LIxvt.Ixvdxdt=Ixv.,τL20,L2−IxfL20,L2,(30)−∫0tht−s.vxxx,sds,Ix2vtL2Qτ=−∫0τ∫0L∫0tht−s.vxxx,sdsIx2vtdxdt=−∫0τ∫0tht−s.vxx,sds.Ix2vt0Ldxdt+∫0τ∫0L∫0tht−s.vxx,sdsIxvtdxdt=∫0τ∫0tht−s.vx,sdsIxvt0Ldxdt−∫0τ∫0L∫0tht−s.vx,sds.vtdxdt=−∫0L∫0tht−s.vx,sdsvdx0τ+∫0τ∫0Lh0v2dxdt+∫0τ∫0L∫0th′t−s.vx,sdsvx,tdxdt=−∫0L∫0τhτ−s.vx,sdsvx,τdx+h0.∥v∥L2Qτ2+∫0τ∫0L∫0th′t−s.vx,sdsvdxdt.Substituting equalities (22)-(30) into (21), we obtain
(31)ρ12∥Ixut.,τ∥L20,L2+κ12∥u.,τ∥L20,L2+ρ22∥Ixvt.,τ∥L20,L2+κ22∥v.,τ∥L20,L2+κ12∥Ixv.,τ∥L20,L2+h0∥v∥L2Qτ2=ρ12∥Ixψ∥L20,L2+κ12∥φ∥L20,L2+ρ22∥Ixg∥L20,L2+κ22∥f∥L20,L2+κ12∥Ixf∥L20,L2+κ1∫0τ∫0LvIxutdxdt−κ1∫0τ∫0LuIxvtdxdt+∫0L∫0τhτ−svx,sdsvx,τdx−∫0τ∫0L∫0th′t−s.vx,sdsvx,tdxdt−∫0τ∫0LFIx2utdxdt−∫0τ∫0LGIx2vtdxdt.By using the Cauchyε -inequality, the last six terms in the right-hand side of (31) can be estimated as follows:
(32)κ1∫0τ∫0LvIxutdxdt≤κ1ε12∫0τ∫0Lv2dxdt+κ12ε1∫0τ∫0LIxut2dxdt,(33)−κ1∫0τ∫0LuIxvtdxdt≤κ1ε22∫0τ∫0Lu2dxdt+κ12ε2∫0τ∫0LIxvt2dxdt,(34)∫0L∫0τhτ−svx,sdsvx,τdx≤ε32∫0Lv2x,τdx+12ε3∫0L∫0τhτ−svx,sds2dx≤ε32∫0Lv2x,τdx+12ε3∫0L∫0τh2τ−sds∫0τv2x,sdsdx≤ε32∫0Lv2x,τdx+T2ε3sup0≤τ≤Th2τ∫0L∫0τv2dxdt,(35)−∫0τ∫0L∫0th′t−s.vx,sdsvdxdt≤ε42∫0τ∫0Lv2dxdt+12ε4∫0τ∫0L∫0th′2t−sds∫0tv2x,sdsdxdt≤ε42∫0τ∫0Lv2dxdt+T2ε4sup0≤t≤Th′2t∫0τ∫0L∫0tv2x,sdsdxdt=ε42∫0τ∫0Lv2dxdt+T2ε4sup0≤t≤Th′2t∫0Lt∫0tv2x,sds0τ−∫0τtv2dtdx=ε42∫0τ∫0Lv2dxdt+T2ε4sup0≤t≤Th′2t∫0L∫0ττ−tv2x,tdtdx≤ε42∫0τ∫0Lv2dxdt+T22ε4sup0≤t≤Th′2t∫0L∫0τv2dtdx=ε42+T22ε4sup0≤t≤Th′2t∫0τ∫0Lv2dxdt,(36)∫0τ∫0LFIx2utdxdt≤ε52∥F∥L2Qτ2+12ε5∥Ix2ut∥L2Qτ2≤ε52∥F∥L2Qτ2+L4ε5∥Ixut∥L2Qτ2,(37)∫0τ∫0LGIx2vtdxdt≤ε62∥G∥L2Qτ2+12ε6∥Ix2vt∥L2Qτ2≤ε62∥G∥L2Qτ2+L4ε6∥Ixvt∥L2Qτ2.If we let(38)ε1=ε2=ε4=ε5=ε6=1,ε3=κ22,we then infer from (31)-(37) that
(39)ρ12∥Ixut.,τ∥L20,L2+κ12∥u.,τ∥L20,L2+ρ22∥Ixvt.,τ∥L20,L2+κ24∥v.,τ∥L20,L2+κ12∥Ixv.,τ∥L20,L2+h0∥v∥L2Qτ2+∥Ixut∥L2Qτ2≤ρ22∥Ixg∥L20,L2+κ22∥f∥L20,L2+κ12∥Ixf∥L20,L2+ρ12∥Ixψ∥L20,L2+κ12∥φ∥L20,L2+L4+κ12∥Ixvt∥L2Qτ2+L4+κ12∥Ixut∥L2Qτ2+κ12∥u∥L2Qτ2+κ12+Tκ2sup0≤t≤Th2t+12+T24sup0≤t≤Th′2t∥v∥L2Qτ2+12∥F∥L2Qτ2+12∥G∥L2Qτ2.By discarding the last three terms on the left-hand side of (39) and using the poincare′ type inequality [27], we have
(40)∥u.,τ∥L20,L2+∥v.,τ∥L20,L2+∥Ixut.,τ∥L20,L2+∥Ixvt.,τ∥L20,L2≤D∥ψ∥L20,L2+∥φ∥L20,L2+∥g∥L20,L2+∥f∥L20,L2+∥F∥L2Qτ2+∥G∥L2Qτ2+∥u∥L2Qτ2+∥v∥L2Qτ2+∥Ixut∥L2Qτ2+∥Ixvt∥L2Qτ2,where
(41)D=maxρ1L/4,ρ2L/4,κ1/2+κ1/κ2sup0≤t≤Th2t+1/2+T2/4sup0≤t≤Th′2t,L/4+κ1/2,κ2/2+κ1L/4ρ1/2,ρ1/2,κ1/2,κ2/4Application of Gronwall’s lemma (see [28]) to (40) gives
(42)∥Ixut.,τ∥L20,L2+∥u.,τ∥L20,L2+∥Ixvt.,τ∥L20,L2+∥v.,τ∥L20,L2≤eDτ∥ψ∥L20,L2+∥φ∥L20,L2+∥f∥L20,L2+∥g∥L20,L2+∥F∥L2Qτ2+∥G∥L2Qτ2≤eDT∥ψ∥L20,L2+∥φ∥L20,L2+∥f∥L20,L2+∥g∥L20,L2+∥F∥L2QT2+∥G∥L2QT2.The right-hand side of inequality (42) does not depend on τ. By taking the supremum with respect to τ over 0,T, we get
(43)ut.,tCJ¯;B210,L+vt.,tCJ¯;B210,L+∥u.,t∥CJ¯;L20,L2+∥v.,t∥CJ¯;L20,L2≤DeDT∥ψ∥L20,L2+∥φ∥L20,L2+∥f∥L20,L2+∥g∥L20,L2+∥F∥L2QT2+∥G∥L2QT2.Then, the a priori estimate (19) follows with C=DeDT.At the moment, we do not have any information about the rangeRA of the operator A except that RA⊂E, and we must extend A so that inequality (43) holds for the extension and its range is the whole space E. In this regard, we prove the following.Proposition 2.
The unbounded operatorA:B→E admits a closure A¯ with domain of definition DA¯.Proof.
LetUn=un,vn∈DA be a sequence such that
(44)Un=un,vn→n→∞0,0inB,(45)AUn=L1un,vn,L2un,vn→n→∞H=H1×H2=F,φ,ψ,G,f,ginE,
where(46)L1un,vn=L1un,vn,ℓ1un,ℓ2un,L2un,vn=L2un,vn,ℓ1vn,ℓ2vn,H1=F,φ,ψ,H2=G,f,g.
Then, we must show thatH1=0 and H2=0. That is, F=0,φ=0,ψ=0,G=0,f=0, and g=0.Equality ((44)) implies that
(47)Un=un,vn→n→∞0,0inD′QT×D′QT,where D′QT is the space of distributions on QT. By the continuity of derivation of D′QT×D′QT→D′QT×D′QT, then (47) implies
(48)limn→∞ρ1untt−κ1unx+vnx+unt=0,limn→∞ρ2vntt−κ2vnxx+κ1unx+vn+∫0tht−svnxxx,sds=0,in D′QT×D′QT. Then, from (45) it follows that
(49)limn→∞ρ1untt−κ1unx+vnx+unt=F,limn→∞ρ2vntt−κ2vnxx+κ1unx+vn+∫0tht−svnxxx,sds=G,in L2QT×L2QT. Therefore,
(50)limn→∞ρ1untt−κ1unx+vnx+unt=F,limn→∞ρ2vntt−κ2vnxx+κ1unx+vn+∫0tht−svnxxx,sds=G,in D′QT×D′QT. By virtue of the uniqueness of the limit in D′QT, the identities (48) and (50) lead to F=0, and G=0.Similarly, we have from (45)
(51)un.,0,vn.,0→n→∞φ,finL20,L×L20,L.We observe from (44) and the obvious inequalities
(52)∥un.,0∥L20,L≤∥un∥B,∀n∈ℕ,∥vn.,0∥L20,L≤∥vn∥B,∀n∈ℕ,that
(53)limn→∞un.,0=0,inL20,L,limn→∞vn.,0=0,inL20,L.We conclude from (51) and (53) and the uniqueness of the limit in L20,L that φ=0, and f=0. In the same manner, we show that ψ=0 and g=0.Definition 3.
The solution of the equationA¯U≪H=F,φ,ψ,G,f,g is called a strong solution of problems (5), (9), and (10).The energy inequality (19) can be extended to
(54)∥U∥B2≤C∥A¯U∥E2,∀U∈DA¯.The previous a priori bound shows that the operatorA¯ is injective and that A¯−1 is continuous from the range RA¯ onto B from which we assert that if a strong solution of problems (5), (9), and (10) exists, it is unique and depends continuously on the initial data φ,ψ,f,g and the free terms F and G.Corollary 4.
The setRA¯⊂E is closed and RA¯=RA¯.


## 4. Solvability of the Posed Problem

Here is the main result of the paper.Theorem 5.
Problems (5), (9), and (10) admit a unique strong solution satisfying
(55)U=u,v∈CJ¯;L20,L,Ut=ut,vt∈CJ¯;B210,L.Moreover, the solutionU=u,v and its time derivative Ut=ut,vt depend continuously on the data F,φ,ψ,G,f,g, that is,
(56)∥u.,t∥CJ¯;L20,L2+∥v.,t∥CJ¯;L20,L2+ut.,tCJ¯;B210,L2+vt.,tCJ¯;B210,L2≤DeDT∥ψ∥L20,L2+∥φ∥L20,L2+∥f∥L20,L2+∥g∥L20,L2+∥F∥L2QT2+∥G∥L2QT2.Proof.
It follows from Corollary4 that in order to prove the existence of the strongly generalized solution of problems (5), (9), and (10), it is sufficient to show that the range RAof the operator A is everywhere dense in the space E; that is, the operator A is injective. To this end, we first prove the density in the following special case.Theorem 6.
If for some functionW=ω1,ω2∈L2QT2 and for elements U∈D0A=U:U∈DAandliu=liv=0,i=1,2, we have
(57)L1u,v,ω1L2QT+L2u,v,ω2L2QT=0,then W=0 a.e in QT.Proof.
Since relation (57) holds for any element of D0A, we take an element U=u,v with special form given by
(58)U=u,v=0,0,if0≤t≤s,∫stτ−tuττx,τdτ,∫stvτx,τdτ,ifs≤t≤T,and consider the system
(59)E1xt=Ix2uttxt=∫Ttω1x,τdτ,E2xt=Ix2vttxt=∫Ttω2x,τdτ..
It follows from the above relations that(60)Ix2utttxt=ω1x,t,Ix2vtttx,t=ω2x,t.Lemma 7.
The functionW=ω1,ω2, defined by (60), belongs to L2QT2.Proof (of Lemma).
We use thet−averaging operator ρε introduced in [32]. By applying the operators ρε and ∂/∂t to the first equation in (59), we obtain
(61)∂∂tIx2utt=∂∂tIx2utt−ρεIx2utt+∂∂tρεE1xt,then
(62)∂∂tIx2uttL2QT2≤2∂∂tIx2utt−ρεIx2uttL2QT2+2∂∂tρεE1xtL2QT2.Using thet−averaging operator ρε properties, we infer from the above inequality that
(63)∂∂tIx2uttL2QT2≤2∂∂tρεE1xtL2QT2.Sinceρεu→u as ε→0 in L2QT and the norm of ∂/∂tIx2utt in L2QT is bounded, we conclude that ω1∈L2QT. Similarly, applying ∂/∂tρε to the second equation in (59), we conclude that ω2∈L2QT. Consequently, W=ω1,ω2∈L2QT2.We now continue the proof of Theorem6. Replacing the functions ω1 and ω2 given by (60) in (57), we obtain
(64)ρ1utt,Ix2utttL2QT−κ1uxx,Ix2utttL2QT−κ1vx,Ix2utttL2QT+ut,Ix2utttL2QT+ρ2vtt,Ix2vtttL2QT−κ2vxx,Ix2vtttL2QT+κ1ux,Ix2vtttL2QT+κ1v,Ix2vtttL2QT+∫0tht−svxxx,sds,Ix2vtttL2QT=0.Invoking the boundary integral conditions and carrying out appropriate integrations by parts of each term, we have(65)ρ1utt,Ix2utttL2QT=∫0T∫0Lρ1uttIx2utttdxdt=∫sT∫0Lρ1uttIx2utttdxdt=ρ12∥Ixutt.,s∥L20,L2,(66)ρ2vtt,Ix2vtttL2QT=ρ22∥Ixvtt.,s∥L20,L2,(67)ut,Ix2utttL2QT=∥Ixutt∥L2Qs2,(68)−κ1uxx,Ix2utttL2QT=κ1uxx,Ix2utttL2Qs=−κ1ux,IxutttL2Qs=κ1u,utttL2Qs=−κ1ut,uttL2Qs=κ12∥ut.,T∥L20,L2,(69)−κ2vxx,Ix2vtttL2QT=κ22∥vt.,T∥L20,L2,(70)−κ1vx,Ix2utttL2QT=−κ1vx,Ix2utttL2Qs=κ1v,IxutttL2Qs=−κ1vt,IxuttL2Qs,(71)κ1ux,Ix2vtttL2QT=κ1ut,IxvttL2Qs,(72)∫0tht−ηvxxx,ηdη,Ix2vtttL2QT=∫0tht−ηvxxx,ηdη,Ix2vtttL2Qs=−∫0tht−ηvxx,ηdη,IxvtttL2Qs=∫0tht−ηvx,ηdη,vtttL2Qs=−h0∫sT∫0Lvttvdxdt−∫sT∫0Lvtt∫0th′t−ηvx,ηdηdxdt=−h0∫0LvvtsTdx+h0∫sT∫0Lvt2dxdt−∫0L∫0th′t−ηvx,ηdηvtsTdx+h′0∫sT∫0Lvtvdxdt+∫sT∫0Lvt∫0th′′t−ηvx,ηdηdxdt=−h0∫0Lvx,Tvtx,Tdx+h0∫sT∫0Lvt2dxdt−∫0Lvtx,T∫0Th′T−ηvx,ηdηdx+h′02∫0Lv2x,Tdx+∫sT∫0Lvt∫0th′′t−ηvx,ηdηdxdt,(73)κ1v,Ix2vtttL2QT=κ1∫sT∫0LvIx2vtttdxdt=−κ1∫sT∫0LvtIx2vttdxdt=κ1∫sT∫0LIxvtIxvttdxdt=κ12∫0LIxvt2sTdx=κ12∥Ixvt.,T∥L20,L2.Substitution of equations (65)-(73) into (64) gives
(74)ρ12∥Ixutt.,s∥L20,L2+κ12∥ut.,T∥L20,L2+ρ22∥Ixvtt.,s∥L20,L2+κ22∥vt.,T∥L20,L2+κ12∥Ixvt.,T∥L20,L2+h′02∥v.,T∥L20,L2+h0∥vt∥Qs2=−κ1∫QsutIxvttdxdt+κ1∫QsvtIxuttdxdt+h0∫0Lvt.,Tvx,Tdx+∫0Lvt.,T∫0Th′T−ηvx,ηdηdx−∫Qsvt∫0th′′t−ηvx,ηdηdxdt.By using the Cauchyε -inequality, we estimate each term of the right-hand side of the previous relations to get
(75)κ1∫QsvtIxuttdxdt≤κ1ε12∥vt∥L2Qs2+κ12ε1∥Ixutt∥L2Qs2,(76)−κ1∫QsutIxvttdxdt≤κ1ε22∥ut∥L2Qs2+κ12ε2∥Ixvtt∥L2Qs2,(77)h0∫0Lvtx,Tvd,Tdx≤h0ε32∥vt.,T∥L20,L2+h02ε3∥v.,T∥L20,L2≤h0ε32∥vt.,T∥L20,L2+h02ε3∥v∥L2Qs2+h02ε3∥vt∥L2Qs2,(78)∫0Lvtx,T∫0Th′T−ηvx,ηdηdx≤ε42∫0Lvt2x,Tdx+12ε4∫0L∫0Th′2T−ηdη∫0Tv2x,ηdηdx≤ε42∫0Lvt2x,Tdx+Tsup0≤t≤Th′2t2ε4∫0L∫sTv2dtdx,(79)−∫sT∫0L∫0th′′t−ηvx,ηdηvtx,tdxdt≤ε52∫sT∫0Lvt2dxdt+12ε5∫sT∫0L∫0th′′t−ηvx,ηdη2dx≤ε52∫sT∫0Lvt2dx+12ε5∫sT∫0L∫0th′′2t−ηdη1/2∫0tv2x,ηdη1/22dx⏟usingHolder′sinequality≤ε52∥vt∥L2Qs2+12ε5∫sT∫0L∫0th′′2t−ηdη∫0tv2x,ηdηdxdt≤ε52∥vt∥L2Qs2+Tsup0≤t≤Th′′2t2ε5∫sT∫0L∫0tv2x,ηdηdxdt=ε52∥vt∥L2Qs2+Tsup0≤t≤Th′′2t2ε5∫0L∫0TT−tv2dtdx≤ε52∥vt∥L2Qs2+T2sup0≤t≤Th′′2t2ε5∫0L∫sTv2dtdx.By combining equality (74) and inequalities (75)-(79) and taking
(80)ε1=2h0κ1,ε2=12,ε3=κ22h0,ε4=κ24,ε5=1,we obtain
(81)ρ12∥Ixutt.,s∥L20,L2+κ12∥ut.,T∥L20,L2+ρ22∥Ixvtt.,s∥L20,L2+κ28∥vt.,T∥L20,L2+κ22∥Ixvt.,T∥L20,L2+h′02∥v.,T∥L20,L2≤h20κ2+12∥vt∥L2Qs2+κ124h0∥Ixutt∥L2Qs2+κ14∥ut∥L2Qs2+κ1∥Ixvtt∥L2Qs2+2Tsup0≤t≤Th′2tκ2+h20κ2+T2sup0≤t≤Th′′2t2∥v∥L2Qs2.By discarding the termsκ2/2∥Ixvtx,T∥L20,L2, ∥vx,T∥L20,L2 from the left-hand side of (81) and using the Frederick’s inequality for the norm of v obtained from the norm of vt, it follows that
(82)∥Ixutt.,s∥L20,L2+∥Ixvtt.,s∥L20,L2+∥ut.,T∥L20,L2+∥vt.,T∥L20,L2≤C∗∫sT∥Ixutt∥L20,L2+∥Ixvtt∥L20,L2+∥ut∥L20,L2+∥vt∥L20,L2dt,where
(83)C∗=max1/2+h20/κ2,κ1,M∗2Tsup0≤t≤Th′2t/κ2+T2sup0≤t≤Th′′2t/2+h20/κ2minρ1/2,κ1/2,ρ2/2,κ2/8,h′0/2,and M∗ is the Frederick’s constant.Inequality (82) is important and fundamental in the proof; to use it, we introduce the new functions ξ,μ defined by
(84)μx,t=∫tTvττx,τdτ,ξx,t=∫tTuττx,τdτ.Then,μx,s=vtx,T,vtx,t=μx,s−μx,t and ξx,s=utx,T,utx,t=ξx,s−ξx,t, and we have
(85)∫sT∥ut∥L20,L2dt≤2∫sT∥ξx,t∥L20,L2dt+2T−sξx,sL20,L,∫sT∥vt∥L20,L2dt≤2∫sT∥μx,t∥L20,L2dt+2T−sμx,sL20,L.Consequently, inequality (82) reduces to
(86)∥Ixutt.,s∥L20,L2+1−2C∗T−s∥ξ.,s∥L20,L2+∥Ixvtt.,s∥L20,L2+1−2C∗T−s∥μ.,s∥L20,L2≤2C∗∫sT∥Ixutt.,t∥L20,L2+∥ξ.,t∥L20,L2+∥Ixvtt.,t∥L20,L2+∥μ.,t∥L20,L2dt.Chooses0≥0 such that s∈T−s0,T. and 2C∗T−s0=1l2; then, inequality (86) implies
(87)∥Ixutt.,s∥L20,L2+∥ξ.,s∥L20,L2+∥Ixvtt.,s∥L20,L2+∥μ.,s∥L20,L2≤4C∗∫sT∥Ixutt.,t∥L20,L2+∥ξ.,t∥L20,L2+∥Ixvtt.,t∥L20,L2+∥μ.,t∥L20,L2dt,for all s∈T−s0,T. If in (86) we let
(88)Λs=∫sT∥Ixutt.,t∥L20,L2+∥ξ.,t∥L20,L2+∥Ixvtt.,t∥L20,L2+∥μ.,t∥L20,L2dt,then it follows from (87) that
(89)−dΛs2ds≤C∗Λs.Thus,(90)−ddsΛse2C∗s≤0.It then follows from inequality (90) that ΛT=0, and hence, W=ω1,ω2=0 almost everywhere in QT−s0. Proceeding in this way step by step along a rectangle of side s0, we prove that W≡0 almost everywhere in QT.We now consider the general case for density.SinceE is a Hilbert space, then RA¯=E is equivalent to the orthogonality of the vector N≪N1,N2=ω1,ω3,ω4,ω2,ω5,ω6∈E to the set RA, that is, if and only if the relation
(91)AU,NE=L1u,v,L2u,v,N1,N2E=L1u,v,ℓ1u,ℓ2u,L2u,v,ℓ1v,ℓ2v,ω1,ω2,ω3,ω4,ω5,ω6E=L1u,v,ω1L2QT+ℓ1u,ω2L20,L+ℓ2u,ω3L20,L+L2u,v,ω4L2QT+ℓ1v,ω5L20,L+ℓ2v,ω6L20,L=0,where U runs over the space B and N∈E, which implies that N≪N1,N2=0,0. That is, ω1=ω2=ω3=ω4=ω5=ω6=0.LetU∈D0A; then, equation (91) becomes
(92)L1u,v,ω1L2QT+L1u,v,ω4L2QT=0.Hence, by virtue of Theorem6, it follows from (92) that ω1=ω4=0. Consequently, equation (91) takes the form
(93)ℓ1u,ω2L20,L+ℓ2u,ω3L20,L+ℓ1v,ω5L20,L+ℓ2v,ω6L20,L=0.Since the four terms in (93) vanish independently and since the ranges Rℓ1,Rℓ2 of the trace operators ℓ1,ℓ2 are everywhere dense in the space L20,L, then it follows from (93) that ω2=ω3=ω5=ω6=0. Consequently, N=0., that is, RA⊥=0. Thus, RA¯=E.


## 5. Conclusion

In this article, we proved the well posedness of a nonhomogeneous Timoshenko system with a viscoelastic damping term. The coupled two hyperbolic equations were associated with initial conditions and nonlocal boundary conditions. The proofs of the results are mainly based on some energy and a priori estimates and on some density arguments. The method uses functional analysis tools such as operator theory and density arguments. It is found that the method is efficient and powerful for solving initial boundary value problems with nonlocal constraints. The a priori estimate for the solution can be provided by constructing suitable multiplicators and from which it is also possible to establish the solvability of the stated problem. We note, here, that no previous works were done for Timoshenko systems with nonlocal conditions of integral type.



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*Source: 6674060-2021-02-28.xml*