MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation92917610.1155/2011/929176929176Research ArticleOn Spatial Evolution of the Solution of a Nonstandard Problem in Linear Thermo-Microstretch ElasticityBulgariuEmilianLuongoAngelo1Faculty of MathematicsAlexandra Ioan Cuza University700506 IasiRomaniauaic.ro20112382011201103052011170620112011Copyright © 2011 Emilian Bulgariu.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An anisotropic and nonhomogeneous compressible linear thermo-microstretch elastic cylinder is subject to zero body loads and heat supply and zero lateral specific boundary conditions. The motion is induced by a time-dependent displacement, microrotation, microstretch, and temperature variation specified pointwise over the base. Further, the motion is constrained such that the displacement, microrotation, microstretch and temperature variation and their derivatives with respect to time at points in the cylinder and at a prescribed time are given in proportion to, but not identical with, their respective initial values. Two different cases for these proportional constants are treated. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show that the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay.

1. Introduction

The theory of micromorphic bodies was introduced by Eringen [1, 2] in order to adequately describe the behavior of materials that have internal structure, such as liquid crystals, blood flow, polymeric substances, and porous materials. Deformation of particles that compose the material contributes to the macroscopic behavior of the body. Eringen also developed the theory of microstretch elastic solids  which is a generalization of the micropolar theory . The particles of the solid with microstretch can expand and contract independent of translations and the rotations which they execute. Later, Eringen developed the theory of thermo-microstretch elastic solids . For this class of materials, De Cicco and Nappa  derived the equations of the linear theory of thermo-microstretch elastic solids with the help of an entropy production inequality proposed by Green and Laws . Bofill and Quintanilla  studied existence and uniqueness results. In the case of semi-infinite cylinders with the boundary lateral surface at null temperature, Quintanilla  established a spatial decay estimate controlled by an exponential of a polynomial of second degree. The spatial and temporal behavior of thermoelastodynamic processes for microstretch continuum materials was studied by Ciarletta and Scalia .

The class of the nonstandard problems attracted the attention of many researchers in the last two decades: Ames, Payne, Knops, Song, Ciarletta, Chiriţă, Quintanilla, Straughan, Passarella, and others. Knops and Payne  studied spatial behavior for the motion of a (semi-infinite) cylinder composed of a nonhomogeneous anisotropic linear elastic material and subject to zero body force and zero lateral boundary conditions. The initial displacement and velocity are not prescribed, nor is the asymptotic behavior at large axial distance. It is prescribed a proportion between displacement and velocity at a given time and, respectively, their initial values. Similar problems were studied by Chiriţă and Ciarletta  for the theory of linear thermoelasticity without energy dissipation and by Bulgariu  for the theory of elasticity with voids.

We consider a cylinder occupied by an anisotropic nonhomogeneous compressible linear thermo-microstretch elastic material, which is subject to null supply terms and null lateral boundary conditions. The internal energy density per unit of initial volume is assumed to be positive definite, and the constitutive coefficients are assumed bounded from above. Initial data are not prescribed, neither is the asymptotic behavior at large axial distance. We establish decay and growth exponential estimates with respect to axial variable for an integral of cross-sectional energy.

The problem studied in this paper finds application in geology and structural engineering. In  we have the example of a pile driven into a rigid foundation that prevents movement of the lateral boundary. The time-dependent displacement, microrotation, microstretch, and variation of temperature prescribed over the excited end constrains the motion such that the displacement, microrotation, microstretch, and variation of temperature and their derivatives with respect to time at points in the cylinder and at a prescribed time are given in proportion to, but not identical with, their respective initial values. It is desired to predict the deformation at each cross-section of the pile in terms of the base displacement, microrotation, microstretch, and variation of temperature.

2. Notation and Basic Formulation

Consider a prismatic cylinder Ω3 whose bounded uniform cross-section D2 has piecewise continuously differentiable boundary D.

The standard convention of summation over repeated suffixes is adopted, and a subscript comma denotes the spatial partial differentiation with respect to the corresponding cartesian coordinate and a superposed dot denotes differentiation with respect to time. Greek subscripts vary over {1,2}, and Latin subscripts vary over {1,2,3}. The letter s is reserved for use as a time integration variable.

With respect to the chosen Cartesian coordinates, a partial volume of the cylinder will be denoted byΩ(z1,z2)={xΩ:z1x3z2}, and, for semi-infinite cylinder, it is convenient to introduce the abbreviationsΩ(z)={xΩ:zx3},Ω0(z)={xΩ:zx3  at  time  t=0}. For the cylinder of finite length L (if the cylinder is semi-finite we take L=), Ω is equivalent with Ω(0,L), while Ω and Ω(0) are equivalent for the semi-infinite cylinder. To be more explicit, we will employ the notation D(x3,t) to indicate that respective quantities are to be evaluated at time t over the cross-section whose distance from the origin is x3.

In this paper, we consider the theory of thermo-microstretch elastic solids. The equations of this theory are 

the evolutive equations: tji,j+fi=ρüi,mji,j+ɛirstrs+gi=Iijφ̈j,πi,i-σ+h=Jψ̈,ρT0η̇=qi,i+s        in  Ω×(0,T),

the constitutive equations: tij=Aijrsers+Bijrsκrs+Dijrγr+Aijψ-βijθ,mij=Brsijers+Cijrsκrs+Eijrγr+Bijψ-Cijθ,3πi=Drsiers+Ersiκrs+Dijγj+diψ-ξiθ,3σ=Arsers+Brsκrs+diγi+mψ-ζθ,ρη=βrsers+Crsκrs+ξiγi+ζψ+aθ,qi=kijθ,jin  Ω¯×[0,T),

the geometric relations

eij=uj,i+εjikφk,κij=φj,i,γi=ψ,i,on  Ω¯. In the above equations we have used the following notations: tij is the stress tensor, mij is the couple stress tensor, πi is the microstress vector, σ is the scalar microstress function, η is the specific entropy, ρ is the mass density (mass in the reference configuration), fi is the body force, gi is the body couple, h is the (scalar) body load, s is the heat source density, qi is the heat flux vector, Iij is the microinertia tensor, J is the microstretch inertia, and ɛijk is the alternating symbol. The variables of this theory are as follows: ui the components of the displacement vector, φi the components of the microrotation vector, ψ the microstretch function, and θ the variation of temperature from the uniform reference absolute temperature T0.

The constitutive coefficients and Iij are prescribed functions of the spatial variable with the following symmetries:Aijrs=Arsij,Cijrs=Crsij,Dij=Dji,kij=kji,Iij=Iji. Moreover we havekijθ,iθ,j0,Iijφ,iφ,j0.

We assume that ρ,  Iij,  J and the constitutive coefficients are continuous and bounded fields on the closure Ω¯. We also assume that the constitutive coefficients are continuous differentiable functions on Ω¯ andρ(x)ρ0>0,a(x)a0>0,J(x)J0>0,I(x)I0>0, where I(x) denote the minimum eigenvalue of Iij(x) and ρ0,  a0,  J0,  I0 are constants.

By taking into account that kij is a positive definite tensor, we havekmθ,iθ,ikijθ,iθ,jkMθ,iθ,i, where km and kM are related to the minimum and the maximum eigenvalue (conductivity moduli) for kij. By using the Schwarz's inequality we haveqiqikMkijθ,iθ,j.

In what follows we denote with 𝒫 the nonstandard problem structured by equations (2.3)–(2.5) with null supply terms, supplemented by the lateral boundary conditionsu̇itαinα=0,φ̇imαinα=0,ψ̇παnα=0,θqαnα=0,(x,t)(D×[0,L])×[0,T], conditions on the baseui(x,t)=ai(xα,t),φi(x,t)=bi(xα,t),ψ(x,t)=c(xα,t),θ(x,t)=τ(xα,t),(x,t)D(0)×[0,T], and the final values at time T of the displacement, microrotation, microstretch, and variation of temperature and their derivatives with respect to time are proportional to their initial values, that is,ui(x,T)=λui(x,0),φi(x,T)=λφi(x,0),ψ(x,T)=λψ(x,0),θ(x,T)=νθ(x,0)for  xΩ,u̇i(x,T)=αu̇i(x,0),φ̇i(x,T)=μφ̇i(x,0),ψ̇(x,T)=βψ̇(x,0),xΩ, where n is the unit outward normal on D,  ai(xα,t),  ai(xα,t),  c(xα,t), and τ(xα,t) are prescribed differentiable functions compatible with the initial/final data and the lateral boundary data. The constants α,  μ,  λ,  β, and ν are prescribed and satisfy the conditions|α|>1,|μ|>1,|β|>1,|λ|>1|ν|>1.

The initial displacement, microrotation, microstretch, and variation of temperature and their derivatives with respect to time at points in the cylinder are not prescribed. The conditions specified on the end D(L) for a finite cylinder, or at asymptotically large axial distance for the semi-infinite cylinder, are also not prescribed.

We will use the notations ϰ={ψij,χij,νi,ϕ} and ϰ(α)={ψij(α),χij(α),νi(α),ϕ(α)}, α=1,2. We assume that the internal energy density per unit of volume is a positive definite quadratic form, therefore we can writemψijψij+mχijχij+mνiνi+mϕ22W(ϰ)Mψijψij+Mχijχij+Mνiνi+Mϕ2,for  all  ϰ, where m,  M,  m,  M,  m,  M,  m,  M are positive constants related to the minimum and, respectively, maximum eigenvalues of the positive definite quadratic form 2W(ϰ)=Aijrsψijψrs+Cijrsχijχrs+Dijνiνj+mϕ2+2Bijrsψijχrs+2Dijrψijνr+2Eijrχijνr+2Aijψijϕ+2Bijχijϕ+2diνiϕ.

By dedublation of this quadratic form we have2E(ϰ(1),ϰ(2))=Aijrsψij(1)ψrs(2)+Cijrsχij(1)χrs(2)+Dijνi(1)νj(2)+mϕ(1)ϕ(2)+Bijrs[ψij(1)χrs(2)+ψij(2)χrs(1)]+Dijr[ψij(1)νr(2)+ψij(2)νr(1)]+Eijr[χij(1)νr(2)+χij(2)νr(1)]+Aij[ψij(1)ϕ(2)+ψij(2)ϕ(1)]+Bij[χij(1)ϕ(2)+χij(2)ϕ(1)]+di[νi(1)ϕ(2)+νi(2)ϕ(1)]. We can remark that (ϰ,ϰ)=W(ϰ). By the Cauchy-Schwarz inequality, we haveE(ϰ(1),ϰ(2))[W(ϰ(1))]1/2[W(ϰ(2))]1/2.

We introduce the following notations: Tij=tij+βijθ, Mij=mij+Cijθ, Πi=3πi+ξiθ, Σ=3σ+ζθ. If, in (2.16), we take ϰ¯={(1/ρ0)Tij,(1/I0)Mij,  (1/J0)Πi,(1/ρ0)Σ}, we obtain2W(ϰ¯)Mρ02TijTij+MI02MijMij+MJ02ΠiΠi+Mρ02Σ2ϖ(1ρ0TijTij+1I0MijMij+1J0ΠiΠi+1ρ0Σ2), where ϖ=max(Mρ0,MI0,MJ0,Mρ0).

From (2.17)–(2.20), we have1ρ0TijTij+1I0MijMij+1J0ΠiΠi+1ρ0Σ2=2E({eij,κij,γi,ψ},{1ρ0Tij,1I0Mij,1J0Πi,1ρ0Σ})[2W]1/2[2W(ϰ¯)]1/2[2ϖW(1ρ0TijTij+1I0MijMij+1J0ΠiΠi+1ρ0Σ2)]1/2, where W=W({eij,κij,γi,ψ}), and consequently we obtain1ρ0TijTij+1I0MijMij+1J0ΠiΠi2ϖW.

By using relations (2.4), (2.5), and (2.17), we obtaintijėij+mijκ̇ij+3πiγ̇i+3σψ̇+ρη̇θ+1T0qiθ,i=Ẇ+1T0kijθ,iθ,j+aθθ̇.

3. A Differential Inequality

The aim of this section is to obtain a differential inequality for an appropriate function related to the cross-sectional energy flux.

We introduce the following function:I(x3)=0TD(x3,s)e-ωs(t3iu̇i+m3iφ̇i+3π3ψ̇+1T0q3θ)dads,0x3L, where ω is a positive parameter at our disposal whose values will be defined later.

By direct differentiation with respect to x3 in (3.1) and by using the evolutive equations (2.3) with null supply terms, we obtain dIdx3(x3)=0TD(x3,s)e-ωs(1T0  ρu̇iüi+Iijφ̇iφ̈j+3Jψ̇ψ̈+ρη̇θ-tαi,αu̇i+t3iu̇i,3-mαi,αφ̇i-ɛjiktjiφ̇k+m3iφ̇i,3-3πα,αψ̇+3σψ̇+3π3ψ̇,3-1T0qα,αθ+1T0q3θ,3)dads, and by using the geometric relations (2.5), we have dIdx3(x3)=0TD(x3,s)e-ωs(1T0qiθ,iρu̇iüi+Iijφ̇iφ̈j+3Jψ̇ψ̈+ρη̇θ+tijėij+mijκ̇ij+3πiγ̇i+3σψ̇+1T0qiθ,i)dads-0TD(x3,s)e-ωs(tαiu̇i+mαiφ̇i+3παψ̇+1T0qαθ),αdads. The divergence theorem and the lateral boundary conditions (2.11) ensure us that the final integral in the right-hand term vanishes. Using relation (2.24), we deduce dIdx3(x3)=120TD(x3,s)e-ωss(ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇ψ̇+2W+aθ2)dads+0TD(x3,s)1T0e-ωskijθ,iθ,jdads.

Finally, the above equation yieldsdIdx3(x3)=E(x3,T)-E(x3,0)+0TωE(x3,s)ds+0TD(x3,s)1T0e-ωskijθ,iθ,jdads, whereE(x3,t)=12D(x3,s)e-ωs(ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇2+2W+aθ2)da,t[0,T]. Therefore, by means of relations (2.13) and (2.14), we havedIdx3(x3)=12(e-ωTα2-1)D(x3,0)ρu̇iu̇ida+12(e-ωTμ2-1)D(x3,0)Iijφ̇iφ̇jda+12(e-ωTβ2-1)D(x3,0)3Jψ̇2da+12(e-ωTλ2-1)D(x3,0)2Wda+12(e-ωTν2-1)D(x3,0)aθ2da+0TωE(x3,s)ds+0TD(x3,s)1T0e-ωskijθ,iθ,jdads.

We will choose the parameter ω so that we have0<χω=12min(e-ωTα2-1,e-ωTμ2-1,e-ωTβ2-1,e-ωTλ2-1,e-ωTν2-1), assumming that ω ranges in the set0<ω<2Tmin(ln|α|,ln|μ|,ln|β|,ln|λ|,ln|ν|) if we suppose that conditions (2.15) hold true.

In this context, we note thatdIdx3(x3)χωD(x3,0)(ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇2+2W+aθ2)da+0TωE(x3,s)ds+0TD(x3,s)1T0e-ωskijθ,iθ,jdads0, in view of assumptions of the positive definitiveness of W and kij. Therefore, we can conclude that I(x3) is a nondecreasing function with respect to x3 on [0,L].

Next, we want to obtain an appropriate estimate for the function I(x3). Using the constitutive equations, the Schwarz's inequality, the arithmetic-mean inequality, and relation (2.23), we obtain the inequality1ρ0tijtij+1I0mijmij+1J03πiπi=(Tijρ0-βijρ0θ)(Tijρ0-βijρ0θ)+(MijI0-CijI0θ)(MijI0-CijI0θ)+(Πi3J0-ξi3J0θ)(Πi3J0-ξi3J0θ)(1+ɛ1)(1ρ0TijTij+1I0MijMij+13J0ΠiΠi)  +(1+1ɛ1)(1ρ0βijβij+1I0CijCij+13J0ξiξi)θ2(1+ɛ1)2ϖW+(1+1ɛ1)M2θ2,ɛ1>0, where M2=maxΩ¯(1ρ0βijβij+1I0CijCij+13J0ξiξi). Using Schwarz's inequality, arithmetic-mean inequality, and relations (2.10) and (3.11), we obtain the estimate|I(x3)|0TD(x3,s)e-ωs{ɛ22[1ρ0t3it3i+1I0m3im3i+3J0π3π3]+ɛ32T0a0q3q3+12ɛ2[ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇2]+12T0ɛ3aθ2}dads0TD(x3,s)e-ωs{ɛ22[1ρ0tijtij+1I0mijmij+3J0πiπi]+ɛ32T0a0qiqi+12ɛ2[ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇2]+12T0ɛ3aθ2}dads0TD(x3,s)e-ωs{1ωɛ2ω2[ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇2]+ɛ2(1+ɛ1)ϖωω22W+[ɛ2M2(1+ɛ1)ωa0ɛ1+1ωT0ɛ3]ω2aθ2+ɛ3kM2a01T0kijθ,iθ,j}dads,ɛ1,ɛ2,ɛ3>0. Further, we equate the coefficients of all energetic terms in the last integral imposing that1ωɛ2=ɛ2(1+ɛ1)ϖω=M2ɛ2(1+ɛ1)ωa0ɛ1+1ωT0ɛ3=ɛ3kM2a0, and hence we haveɛ2=1(1+ɛ1)ϖ,ɛ3=2a0kMω(1+ɛ1)ϖ, whereɛ1=12[-(1-2T0M2+ωkM2a0T0ϖ)+(1-2T0M2+ωkM2a0T0ϖ)2+4M2a0ϖ].

Because we imposed (3.14), we can multiply (3.13) by c=ωɛ2 and obtain c|I(x3)|0TωE(x3,s)ds+0TD(x3,s)1T0e-ωskijθ,iθ,jdads, and from (3.10) we obtain the first-order differential inequalityc|I(x3)|dIdx3(x3),x3[0,L].

4. Spatial Behaviour of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M119"><mml:mi>I</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>

In this section we determine the spatial evolution of the solution of the nonstandard problem 𝒫 by integrating (3.18). We first consider that the cylinder has a finite length (i.e., L<). We have only two possibilities: (a) I(x3)0 for all x3[0,L] or (b) there exists z[0,L] so that I(z)>0. It is easy to see that, if I(L)0, we are in the case (a).

Let us first consider the case (a). Because I(x3) it is a nondecreasing function with respect to x3 on [0,L], we havedIdx3(x3)+cI(x3)0,0x3L, which after integration leads to the following Saint-Venant-type decay estimate:0-I(x3)-I(0)e-cx3,0x3L. For a cylinder of finite length L, we have to prescribe such boundary conditions on the end x3=L that implies I(L)=0, and then we will predict a spatial exponential decay as described in (4.2).

For the case (b), we have 0I(z)I(x3) for zx3L, and hence (3.18) implies the inequality0dIdx3(x3)-cI(x3),zx3L, which, by integrating, yields the following growth estimateI(x3)I(z)ec(x3-z),zx3L.

Let us discuss further the case of a semi-infinite cylinder (i.e., the case when L).

If I(x3)0 for all x3[0,), we obtain that I(x3)0 as x3, and hence relation (3.10) gives the following decay estimate for the weighted total energy:E(x3)-I(0)e-cx3,0x3<, whereE(x3*)=χωΩ0(x3*)(ρu̇iu̇i+Iijφ̇iφ̇j+3Jψ̇2+2W+aθ2)dv+limx30Tx3*x3ωE(ϑ,s)dϑds+0TΩ(x3*)1T0e-ωskijθ,iθ,jdvds,0x3*<.

If there is z[0,) so that I(z)>0, for the semi-infinite cylinder, I(x3) becomes unbounded for x3 and hence (x3) is infinite.

We have established a Phragmén-Lindelöf alternative type for the semi-infinite cylinder.

In this paper we have discussed only the case when |α|>1,  |μ|>1,  |β|>1,  |λ|>1, and |ν|>1. When we take the conditions |α|<1,  |μ|<1,  |β|<1,  |λ|<1 and |ν|<1 we cannot find a suitable bound because there is nonuniqueness in this case (see e.g., Quintanilla and Straughan  for an argument concerning this subject).

In the previous sections we have considered the nonstandard problem 𝒫 in which the proportionality coefficients of displacement, microrotation, microstretch, and variation of temperature with their derivatives with respect to time at the time T and their respective initial values are given by (2.13) and (2.14). We consider a similar nonstandard problem 𝒫* given by (2.3)–(2.5) with null supply terms, and instead of conditions (2.13), we haveui(x,T)=λ1ui(x,0),φi(x,T)=λ2φi(x,0),ψ(x,T)=λ3ψ(x,0),θ(x,T)=νθ(x,0),xΩ, where λi and λj can be different for ij, with conditions (2.14) remaining valid. The problem 𝒫* has the lateral boundary conditionsu̇itαinα=0,φ=0,ψ̇παnα=0,θqαnα=0,(x,t)(D×[0,L])×[0,T] and the conditions on the baseui(x,t)=ai(xα,t),φi(x,t)=bi(xα,t),ψ(x,t)=c(xα,t),θ(x,t)=τ(xα,t),(x,t)D(0)×[0,T].

We are interested in what conditions we would have to take for the constants λ1,λ2, and λ3 so that our study given in the previous sections may follow the same path.

The internal energy density per unit of volume is a positive definite quadratic form and so, for ϰ={eij,κij,γi,ψ} in (2.16), at the moment t=T, we have2D(x3,T)WdamD(x3,T)eijeijda+mD(x3,T)κijκijda+mD(x3,T)γiγida+mD(x3,T)ψ2da=mD(x3,0)(λ12uj,iuj,i+2λ1λ2ɛjikuj,iφk+2λ22φkφk)2da+mλ222D(x3,0)φj,iφj,ida+mλ222D(x3,0)φj,iφj,ida+mλ32D(x3,0)ψ,iψ,ida+mλ32D(x3,0)ψ2da. The condition required in (5.2) that φ=0 on D(x3) gives us the possibility to apply the Poincaré inequalityD(x3,0)φj,iφj,idaδD(x3,0)φiφida, with δ a positive constant. Using the arithmetic-mean inequality, we can deduce2λ1λ2ɛjikuj,iφk-2|λ1uj,i||λ2ɛjikφk|-ɛλ12uj,iuj,i-2ɛλ22φkφk,ɛ>0.

Combining relations (5.5), (5.6), and (5.4), we have2D(x3,T)Wda(1-ɛ)mλ12D(x3,0)uj,iuj,ida+(2m-2mɛ+mδ2)λ22D(x3,0)φiφida+mλ222D(x3,0)φj,iφj,ida+mλ32D(x3,0)ψ,iψ,ida+mλ32D(x3,0)ψ2da. Requiring that (1+mδ4m)-1<ɛ<1, we ensured that the brackets from the right-hand terms in (5.7) are positive.

Using the inequality eijeij=(uj,i+ɛjikφk)2<2uj,iuj,i+4φkφk, from (2.16), at the moment t=0, we have 2D(x3,0)WdaMD(x3,0)eijeijda+MD(x3,0)κijκijda+MD(x3,0)γiγida+MD(x3,0)ψ2da2MD(x3,0)uj,iuj,ida+4MD(x3,0)φkφkda+MD(x3,0)φj,iφj,ida+MD(x3,0)ψ,iψ,ida+MD(x3,0)ψ2da.

Combining relations (5.7) and (5.10), we have12e-ωTD(x3,T)2Wda-12D(x3,0)2W12[(1-ɛ)e-ωTmλ122M-1]2MD(x3,0)uj,iuj,ida+12[(4m-4mɛ-1+mδ)e-ωTλ228M-1]4MD(x3,0)φiφida+12[e-ωTmλ222M-1]MD(x3,0)φj,iφj,ida+12[e-ωTmλ32M-1]MD(x3,0)ψ,iψ,ida+12[e-ωTmλ32M-1]mD(x3,0)ψ2da, and so, instead of relation (3.7), in the case of the problem 𝒫* we obtain dIdx3(x3)12(e-ωTα2-1)D(x3,0)ρu̇iu̇ida+12(e-ωTμ2-1)D(x3,0)Iijφ̇iφ̇jda+12(e-ωTβ2-1)D(x3,0)3Jψ̇2da+12(e-ωTν2-1)D(x3,0)aθ2da+12[(1-ɛ)e-ωTmλ122M-1]2MD(x3,0)uj,iuj,ida+12[(4m-4mɛ-1+mδ)e-ωTλ228M-1]4MD(x3,0)φiφida+12[e-ωTmλ222M-1]MD(x3,0)φj,iφj,ida+12[e-ωTmλ32M-1]MD(x3,0)ψ,iψ,ida+12[e-ωTmλ32M-1]mD(x3,0)ψ2da+0TωE(x3,s)ds+0TD(x3,s)1T0e-ωskijθ,iθ,jdads. We choose ω to ensure that0<χω=12min(e-ωTmλ32M  e-ωTα2-1,e-ωTμ2-1,e-ωTβ2-1,e-ωTν2-1,(1-ɛ)e-ωTmλ122M-1,(4m-4mɛ-1+mδ)e-ωTλ228M-1,e-ωTmλ222M-1,e-ωTmλ32M-1,e-ωTmλ32M-1), and so, the range from which we can take ω is0<ω<1Tlnmin(α2,μ2,β2,ν2,(1-ɛ)mλ122M,(4m-4mɛ-1+mδ)λ228M,mλ222M,mλ32M,mλ32M) if the conditions (2.15)1,2,3,5 hold true and|λ1|>2Mm>1,|λ2|>max(8M4m+mδ,2Mm)>1,|λ3|>max(Mm,Mm)>1.

From relations (5.12) and (5.13), we obtain an inequality similar to (3.10) and so, we can continue like in Sections 3 and 4 to obtain a Saint-Venant type estimate or a Phragmén-Lindelöf-type estimate.

In conclusion, if we replace conditions (2.13) with those given in (5.1), the results obtained in Sections 3 and 4 hold true if in (2.11) we change the lateral boundary condition φ̇mαinα=0 with φ=0 for (x,t)(D×[0,L])×[0,T] and the constants λ1,  λ2,  λ3,  ν,  α,  β and μ must satisfy conditions (2.15)1,2,3,5 and (5.15).

Acknowledgment

The author acknowledges support from the Romanian Ministry of Education and Research through CNCSIS-UEFISCSU, Project PN II-RU TE code 184, no. 86/30.07.2010.

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