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Blood Pressure Monitoring in Cardiovascular Disease

  • While the practice of taking blood pressure readings at the physician’s office continues to be valid, home blood pressure monitoring is being increasingly used to enhance diagnostic accuracy and ensure a more personalized follow-up of patients. In the case of white coat hypertension and resistant arterial hypertension, ambulatory blood pressure monitoring is indispensable. Recent studies attach great importance to nocturnal blood pressure patterns, with a reduction in these becoming a treatment goal, a strategy known as chronotherapy. Home blood pressure monitoring is useful for both diagnosis and follow-up of arterial hypertension. Its use, particularly if combined with other patient-support interventions, serves to improve blood pressure control. Telemonitoring is associated with a decrease in blood pressure values and an increase in patient satisfaction. All studies highlight the importance of patients being supported by a multidisciplinary health care team, since blood pressure telemonitoring with a support team is more effective than simple data telemonitoring. Further studies are called for, especially on the illiterate population, with difficulties posed by technological accessibility and transcriptions into different languages. More cost-effectiveness studies and long-term results are needed to ascertain the true benefit of blood pressure telemonitoring.

    Citation: Carlos Menéndez Villalva, Xose Luis Muiño López-Alvarez, Martín Menéndez Rodríguez, María José Modroño Freire, Olalla Quintairos Veloso, Lea Conde Guede, Sandra Vilchez Dosantos, Manuel Blanco Ramos. Blood Pressure Monitoring in Cardiovascular Disease[J]. AIMS Medical Science, 2017, 4(2): 164-191. doi: 10.3934/medsci.2017.2.164

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  • While the practice of taking blood pressure readings at the physician’s office continues to be valid, home blood pressure monitoring is being increasingly used to enhance diagnostic accuracy and ensure a more personalized follow-up of patients. In the case of white coat hypertension and resistant arterial hypertension, ambulatory blood pressure monitoring is indispensable. Recent studies attach great importance to nocturnal blood pressure patterns, with a reduction in these becoming a treatment goal, a strategy known as chronotherapy. Home blood pressure monitoring is useful for both diagnosis and follow-up of arterial hypertension. Its use, particularly if combined with other patient-support interventions, serves to improve blood pressure control. Telemonitoring is associated with a decrease in blood pressure values and an increase in patient satisfaction. All studies highlight the importance of patients being supported by a multidisciplinary health care team, since blood pressure telemonitoring with a support team is more effective than simple data telemonitoring. Further studies are called for, especially on the illiterate population, with difficulties posed by technological accessibility and transcriptions into different languages. More cost-effectiveness studies and long-term results are needed to ascertain the true benefit of blood pressure telemonitoring.


    Recent research on nonlinear propagation of sound in the case of high amplitude waves has shown that there is a literature on well-grounded partial differential models. (see, e.g., [1,5,7,9,10,11,12,13,16,17,18,20,21,23,24,25,26,27,28,29,30,39,49]). This highly active field of research is being carried out by a wide range of applications such as the medical and industrial use of high intensity ultrasound in lithotripsy, thermotherapy, ultrasound cleaning and ultrasound chemistry. The classical models of nonlinear acoustics are Kuznetsov's equation, the Westervelt equation, and the KZK (Kokhlov-Zabolotskaya-Kuznetsov) equation. For mathematics. Existence and singularity analysis of several types of initial boundary value problems of this second nonlinear order in evolutionary PDEs, we refer (see [19,22,31,32,33,34,35,36,37,38,40,41,42,43,44,45,46,47,48,50,51]). Focusing on the study of sound wave propagation, it should be noted that the MGT equation is one of the nonlinear sound equations describing the propagation of sound waves in gases and liquids. The behavior of sound waves depends strongly on the average property of scattering, scattering, and nonlinear effects. Arises from high-frequency ultrasound (HFU) modeling see ([16,25,41]). The original derivation dates back to [19]. This model is realized through the third order hyperbolic equation

    τuttt+uttc2ΔubΔut=0,

    the unknown function u=u(x,t) denotes the scalar acoustic velocity, c denotes the speed of sound and τ denotes the thermal relaxation. Besides, the coefficient b=βc2 is related to the diffusively of the sound with β(0,τ]. In [19], W Chen and A Palmieri studied the blow-up result for the semilinear Moore- Gibson-Thompson equation with nonlinearity of derivative type in the conservative case defined as following

    βuttt+uttΔuβΔut=|ut|p,xRn,t>0.

    This paper is related to the following works (see [27,46]). Now when we talk about the (MGT) equation with memory term, we have I. Lasieka and X.Wang in [29] studied the exponential decay of energy of the temporally third order (Moore-Gibson-Thompson) equation with a memory term as follow

    τuttt+αuttc2AubAutt0g(ts)Aw(s)ds=0,

    where τ,α,b,c2 are physical parameters and A is a positive self-adjoint operator on a Hilbert space H. The convolution term t0g(ts)Aw(s)ds reflects the memory effects of materials due to viscoelasticity. In [13] I. Lasieka and X. Wang studied the general decay of solution of same problem above. Moore-Gibson-Thompson equation with nonlocal condition is a new posed problem. Existence and uniqueness of the generalized solution are established by using Galerkin method. This problems can be encountered in many scientific domains and many engineering models, see previous works ([20,22,31,32,33,34,35,36,37,42,43,47,48]). Mesloub and Mesloub in [40] have applied the Galerkin method to a higher dimension mixed nonlocal problem for a Boussinesq equation. While, S. Boulaaras, A. Zaraï and A. Draifia investigated the Moore-Gibson-Thompson equation with integral condition in [17]. In motivate by these outcomes, we improve the existence and uniqueness by Galerkin method of the Fourth-Order Equation of Moore-Gibson-Thompson Type with source term and integral condition, this problem was cited by the work of F. Dell'Oro and V. Pata in [24].

    We define the problem as follow

    {utttt+αuttt+βuttϱΔuδΔutγΔutt+t0h(tσ)Δu(σ)dσ=F(x,t),u(x,0)=u0(x), ut(x,0)=u1(x), utt(x,0)=u2(x),uttt(x,0)=u3(x)uη=t0Ωu(ξ,τ)dξdτ,   xΩ. (1.1)

    The convolution term t0h(ts)Δu(s)ds reflects the memory effect of materials due to vicoelasticity, F is a given function and h is the relaxation function satisfying

    (H1) hC1(R+,R+) is a non-increasing function satisfying

    h(0)>0,h0>0/H()<h0. (1.2)

    where H()=0h(s)ds>0, H(t)=t0h(s)ds and h>0,h<0.

    (H2) ζ>0 satisfying

    h(t)ζh(t),t0. (1.3)

    The impartial of this manuscript is to consider the following nonlocal mixed boundary value problem for the Moore-Gibson-Thompson (MGT) equation for all (x;t)QT=(0,T), where ΩRn is a bounded domain with sufficiently smooth boundary Ω. solution of the posed problem.

    We divide this paper into the following: In the second part, some definitions and appropriate spaces have been given. Then, we use the Galerkin's method to prove the existence, and in the fourth part we demonstrate the uniqueness.

    Let V(QT) and W(QT) be the set spaces defined respectively by

    V(QT)={uW12(QT):ut,uttW12(QT),u,uL2h(QT)},

    and

    W(QT)={uV(QT):u(x,T)=0}.L2h(QT)={uV(QT):T0hu(t)dt<},

    where

    hu(t)=Ωt0h(tσ)(u(t)u(σ))2dσdx.

    Consider the equation

    (utttt,v)L2(QT)+α(uttt,v)L2(QT)+β(utt,v)L2(QT)ϱ(Δu,v)L2(QT)δ(Δut,v)L2(QT)γ(Δutt,v)L2(QT)+(Δw,v)L2(QT)=(F,v)L2(QT), (2.1)

    where

    w(x,t)=t0h(tσ)u(x,σ)dσ,

    and (.,.)L2(QT) defend for the inner product in L2(QT), u is supposed to be a solution of (1.1) and vW(QT). Upon using (2.1) and (1.1), we find

    (uttt,vt)L2(QT)α(utt,vt)L2(QT)β(ut,vt)L2(QT)+ϱ(u,v)L2(QT)+δ(ut,v)L2(QT)γ(ut,vt)L2(QT)(w,v)L2(QT)=(F,v)L2(QT)+ϱT0Ωv(t0Ωu(ξ,τ)dξdτ)dsxdt+δT0ΩvΩu(ξ,t)dξdsxdtδT0ΩvΩu0(ξ)dξdsxdtγT0Ωvt(t0Ωuτ(ξ,τ)dξdτ)dsxdt+(u3(x),v(x,0))L2(Ω)+α(u2(x),v(x,0))L2(Ω)+β(u1(x),v(x,0))L2(Ω)γ(Δu1,v(x,0))L2(Ω)T0Ωv(t0Ωw(ξ,τ)dξdτ)dsxdt. (2.2)

    Now, we give two useful inequalities:

    ● Gronwall inequality: If for any tI, we have

    y(t)h(t)+ct0y(s)ds,

    where h(t) and y(t) are two nonnegative integrable functions on the interval I with h(t) non decreasing and c is constant, then

    y(t)h(t)exp(ct).

    ● Trace inequality: When wW21(Ω), we have

    w2L2(Ω)εw2L2(Ω)+l(ε)w2L2(Ω),

    where Ω is a bounded domain in Rn with smooth boundary Ω,  and l(ε) is a positive constant.

    Definition 1. If a function uV(QT) satisfies Eq (2.1) for each vW(QT) is called a generalized solution of problem (1.1).

    Here, by using Galerkin's method, we give the existence of problem (1.1).

    Theorem 1. If u0,u1,u2W12(Ω), u3L2(Ω) and FL2(QT), then there is at least one generalized solution in V(QT) to problem (1.1).

    Proof. Let {Zk(x)}k1 be a fundamental system in W12(Ω), such that

    (Zk,Zl)L2(Ω)=δk,l. 

    First, we will find an approximate solution of the problem (1.1) in the form

    uN(x,t)=Nk=1Ck(t)Zk(x), (3.1)

    where the constants Ck(t) are defined by the conditions

    Ck(t)=(uN(x,t),Zk(x))L2(Ω),    k=1,...,N, (3.2)

    and can be determined from the relations

    (uNtttt,Zl(x))L2(Ω)+α(uNttt,Zl(x))L2(Ω)+β(uNtt,Zl(x))L2(Ω)+ϱ(uN,Zl(x))L2(Ω)+δ(uNt,Zl(x))L2(Ω)+γ(uNtt,Zl(x))L2(Ω)(wN,Zl(x))L2(Ω)=(F(x,t),Zl(x))L2(Ω)+ϱΩZl(x)(t0ΩuN(ξ,τ)dξdτ)dsx+δΩZl(x)(t0ΩuNτ(ξ,τ)dξdτ)dsx+γΩZl(x)(t0ΩuNττ(ξ,τ)dξdτ)dsxΩZl(x)(t0ΩwN(ξ,τ)dξdτ)dsx, (3.3)

    Invoking to (3.1) in (3.3) gives for  l=1,...,N.

    ΩNk=1{Ck(t)Zk(x)Zl(x)+αCk(t)Zk(x)Zl(x)+βCk(t)Zk(x)Zl(x)+ϱCk(t)Zk(x).Zl(x)+δCk(t)Zk(x).Zl(x)+γCk(t)Zk.Zl(t0h(tσ)Ck(σ)dσ)Zk(x).Zl(x)}dx=(F(x,t),Zl(x))L2(Ω)+ϱNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+δNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+γNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτNk=1t0τ0h(τσ)Ck(σ)(ΩZl(x)ΩZk(ξ)dξdsx)dσdτ. (3.4)

    From (3.4) it follows that

    Nk=1Ck(t)(Zk(x),Zl(x))L2(Ω)+αCk(t)(Zk(x),Zl(x))L2(Ω)+βCk(t)(Zk(x),Zl(x))L2(Ω)+ϱCk(t)(Zk,Zl)L2(Ω)+δCk(t)(Zk(x),Zl(x))L2(Ω)+γCk(t)(Zk(x),Zl(x))L2(Ω)(t0h(tσ)Ck(σ)dσ)(Zk,Zl)L2(Ω)}dx=(F(x,t),Zl(x))L2(Ω)+ϱNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+δNk=1t0Ck(τ)(ΩZl(x)ΩZk(ξ)dξdsx)dτ+γNk=1t0(Ck(τ)ΩZl(x)ΩZk(ξ)dξds)dτNk=1t0τ0h(τσ)Ck(σ)(ΩZl(x)ΩZk(ξ)dξdsx)dσdτ,    l=1,...,N. (3.5)

    Let

    (Zk,Zl)L2(Ω)=δkl={1,   k=l0,   kl
    (Zk,Zl)L2(Ω)=γkl,
    ΩZl(x)ΩZk(ξ)dξds=χkl.
    (F(x,t),Zl(x))L2(Ω)=Fl(t).

    Then (3.5) can be written as

    Nk=1Ck(t)δkl+αCk(t)δkl+Ck(t)(βδkl+γγkl)+δCk(t)γkl+ϱCk(t)γklt0(ϱCk(τ)χkl+δCk(τ)χkl+γCk(τ)χklh(tτ)Ck(τ)γkl)t0τ0h(τσ)Ck(σ)dσχkldσdτ=Fl(t). (3.6)

    A differentiation with respect to t (two times), yields

    Nk=1C′′′′′′k(t)δkl+αC′′′′′k(t)δkl+Ck(t)(βδkl+γγkl)+Ck(t)(δγklγχkl)+Ck(t)(ϱγklδχkl)(ϱ+h(0))Ck(t)χkl+h(0)Ck(t)χkl=Fl(t), (3.7)
    {Nk=1[Ck(0)δkl+αCk(0)δkl+Ck(0)(βδkl+γγkl)+δCk(0)γkl+ϱCk(0)γkl]=Fl(0)Ck(0)=(Zk,u0)L2(Ω), Ck(0)=(Zk,u1(x))L2(Ω),Ck(0)=(Zk,u2(x))L2(Ω),Ck(0)=(Zk,u3(x))L2(Ω). (3.8)

    Thus for every n there exists a function uN(x) satisfying (3.3).

    Now, we will demonstrate that the sequence uN is bounded. To do this, we multiply each equation of (3.3) by the appropriate Ck(t) summing over k from 1 to N then integrating the resultant equality with respect to t from 0 to τ, with τT, yields

    (uNtttt,uNt)L2(Qτ)+α(uNttt,uNt)L2(Qτ)+β(uNtt,uNt)L2(Qτ)+ϱ(uN,uNt)L2(Qτ)+δ(uNt,uNt)L2(Qτ)+γ(uNtt,uNt)L2(Qτ)(wN,uNt)L2(Qτ)=(F,uNt)L2(Qτ)+ϱτ0ΩuNt(x,t)(t0ΩuN(ξ,η)dξdη)dsxdt+δτ0ΩuNt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt+γτ0ΩuNt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdtτ0ΩuNt(x,t)(t0ΩwN(ξ,η)dξdη)dsxdt, (3.9)

    after a simplification of the LHS of (3.9), we get

    (uNtttt,uNt)L2(Qτ)=τ0(uNttt,uNtt)L2(Ω)dt+(uNτττ(x,τ),uNτ(x,τ))L2(Ω),(uNttt(x,0),uNt(x,0))L2(Ω),α(uNttt,uNt)L2(Qτ)=α(uNττ(x,τ),uNτ(x,τ))L2(Ω)(uNtt(x,0),uNt(x,0))L2(Ω)ατ0utt(x,t)2L2(Ω)dt,β(uNtt,uNt)L2(Qτ)=β2uNτ(x,τ)2L2(Ω)β2uNt(x,0)2L2(Ω),ϱ(uN,uNt)L2(Qτ)=ϱ2uN(x,τ)2L2(Ω)ϱ2uN(x,0)2L2(Ω),δ(uNt,uNt)L2(Qτ)=δτ0uNt(x,t)2L2(Ω)dt,γ(uNtt,uNt)L2(Qτ)=γ2uNτ(x,τ)2L2(Ω)γ2uNt(x,0)2L2(Ω),(wN,uNt)L2(Qτ)=12huN(τ)12H(τ)uN(x,τ)2L2(Ω)12τ0huN(t)dt+12h(t)uN(x,t)2L2(Ω)dt, (3.10)
    ϱτ0ΩuNt(t0ΩuN(ξ,η)dξdη)dsxdt=ϱΩuN(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uN(x,t)ΩuN(ξ,t)dξdtdsx, (3.11)
    δτ0ΩuNt(t0ΩuNt(ξ,η)dξdη)dsxdt=δΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsxδΩτ0uNt(x,t)ΩuN(ξ,0)dξdtdsx, (3.12)
    γτ0ΩuNt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdt=γτ0ΩuNt(x,t)(ΩuNt(ξ,t)dξ)dsxdtγτ0ΩuNt(x,t)(ΩuNt(ξ,0)dξ)dsxdt. (3.13)
    τ0ΩuNt(t0ΩwN(ξ,η)dξdη)dsxdt=τ0ΩuNt(t0ΩH(η)uN(ξ,η)dξdη)dsxdt+τ0ΩuNt(t0Ω[η0h(ησ)(uN(ξ,η)uN(ξ,σ))dσ]dξdη)dsxdt=ΩuN(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuN(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuN(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuN(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.14)

    Taking into account the equalities (3.10)-(3.14) in (3.9), we obtain

    (uNτττ(x,τ),uNτ(x,τ))L2(Ω)+α(uNττ(x,τ),uNτ(x,τ))L2(Ω)+β2uNτ(x,τ)2L2(Ω)+ϱ2uN(x,τ)2L2(Ω)+γ2uNτ(x,τ)2L2(Ω)+12huN(τ)12H(τ)uN(x,τ)2L2(Ω)=(uNttt(x,0),uNt(x,0))L2(Ω)+α(uNtt(x,0),uNt(x,0))L2(Ω)+ϱ2uN(x,0)2L2(Ω)+γ2uNt(x,0)2L2(Ω)+τ0(uNttt,uNtt)L2(Ω)dt+ατ0utt(x,t)2L2(Ω)dtδτ0uNt(x,t)2L2(Ω)dt+β2uNt(x,0)2L2(Ω)+ϱΩuN(x,τ)τ0ΩuN(ξ,t)dξdtdsx+(F,uNt)L2(Qτ)ϱΩτ0uN(x,t)ΩuN(ξ,t)dξdtdsx+δΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsxδΩτ0uNt(x,t)ΩuN(ξ,0)dξdtdsx+γτ0ΩuNt(x,t)(ΩuNt(ξ,t)dξ)dsxdtγτ0ΩuNt(x,t)(ΩuNt(ξ,0)dξ)dsxdt12τ0huN(t)dt+12h(t)uN(x,t)2L2(Ω)dtΩuN(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuN(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuN(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuN(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.15)

    Now, multiplying each equation of (3.3) by the appropriate Ck(t), add them up from 1 to N and them integrate with respect to t from 0 to τ, with τT, we obtain

    (uNtttt,uNtt)L2(Qτ)+α(uNttt,uNtt)L2(Qτ)+β(uNtt,uNtt)L2(Qτ)+ϱ(uN,uNtt)L2(Qτ)+δ(uNt,uNtt)L2(Qτ)+γ(uNtt,uNtt)L2(Qτ)(wN,uNtt)L2(Qτ)=(F,uNtt)L2(Qτ)+ϱτ0ΩuNtt(x,t)(t0ΩuN(ξ,η)dξdη)dsxdt+δτ0ΩuNtt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt+γτ0ΩuNtt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdtτ0ΩuNtt(x,t)(t0ΩwN(ξ,η)dξdη)dsxdt. (3.16)

    With the same reasoning in (3.9), we find

    (uNtttt,uNtt)L2(Qτ)=τ0uNttt(x,t)2L2(Ω)dt+(uNτττ(x,τ),uNττ(x,τ))L2(Ω)(uNttt(x,0),uNtt(x,0))L2(Ω),α(uNttt,uNtt)L2(Qτ)=α2uNττ(x,τ)2L2(Ω)α2uNtt(x,0)2L2(Ω),β(uNtt,uNtt)L2(Qτ)=βτ0uNtt(x,t)2L2(Ω)dt,ϱ(uN,uNtt)L2(Qτ)=ϱ(uN(x,τ),uNτ(x,τ))L2(Qτ)ϱ(uN(x,0),uNt(x,0))L2(Ω)ϱτ0uNt(x,t)2L2(Ω)dt,δ(uNt,uNtt)L2(Qτ)=δ2uNτ(x,τ)2L2(Ω)δ2uNt(x,0)2L2(Ω),γ(uNtt,uNtt)L2(Qτ)=γτ0uNtt(x,t)2L2(Ω)dt(wN,uNtt)L2(Qτ)=12{huN(τ)+h(τ)uN(x,τ)2L2(Ω)2(wN(τ),uNτ)L2(Ω)}+12τ0huN(t)dt12τ0h(t)uN(x,t)2L2(Ω)dt, (3.17)
    ϱτ0ΩuNtt(t0ΩuN(ξ,η)dξdη)dsxdt=ϱΩuNτ(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsx, (3.18)
    δτ0ΩuNtt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt=δΩuNτ(x,τ)ΩuN(ξ,τ)dξdsxδΩuNτ(x,τ)ΩuN(ξ,0)dξdsxδΩτ0uNt(x,t)ΩuNt(ξ,t)dξdtds, (3.19)
    γτ0ΩuNtt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdt=γΩuNτ(x,τ)ΩuNτ(ξ,τ)dξdsxγΩuNτ(x,τ)ΩuNt(ξ,0)dξdsxγΩτ0uNt(x,t)ΩuNtt(ξ,t)dξdtds, (3.20)
    τ0ΩuNtt(t0ΩwN(ξ,η)dξdη)dsxdt=τ0ΩuNtt(t0ΩH(η)uN(ξ,η)dξdη)dsxdt+τ0ΩuNtt(t0Ω[η0h(ησ)(uN(ξ,η)uN(ξ,σ))dσ]dξdη)dsxdt=ΩuNτ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuNt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuNτ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuNt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.21)

    Upon using (3.17)-(3.21) into (3.16), we have

    (uNτττ(x,τ),uNττ(x,τ))L2(Ω)+α2uNττ(x,τ)2L2(Ω)+δ2uNτ(x,τ)2L2(Ω)+ϱ(uN(x,τ),uNτ(x,τ))L2(Ω)+12h(τ)uN(x,τ)2L2(Ω)12huN(τ)+(wN(τ),uNτ)L2(Ω)=τ0uNttt(x,t)2L2(Ω)dt+(uNttt(x,0),uNtt(x,0))L2(Ω)+α2uNtt(x,0)2L2(Ω)βτ0uNtt(x,t)2L2(Ω)dt+ϱ(uN(x,0),uNt(x,0))L2(Ω)+ϱτ0ut(x,t)2L2(Ω)dt+δ2uNt(x,0)2L2(Ω)+(F,uNtt)L2(Qτ)γτ0uNtt(x,t)2L2(Ω)dt+ϱΩuNτ(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uNt(x,t)ΩuN(ξ,t)dξdtdsx+δΩuNτ(x,τ)ΩuN(ξ,τ)dξdsxδΩuNτ(x,τ)ΩuN(ξ,0)dξdsxδΩτ0uNt(x,t)ΩuNt(ξ,t)dξdtdsx+γΩuNτ(x,τ)ΩuNτ(ξ,τ)dξdsxγΩuNτ(x,τ)ΩuNt(ξ,0)dξdsxγΩτ0uNt(x,t)ΩuNtt(ξ,t)dξdtdsxΩuNτ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuNt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuNτ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuNt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.22)

    Now, multiplying each equation of (3.3) by the appropriate Ck(t), add them up from 1 to N and them integrate with respect to t from 0 to τ, with τT, we obtain

    (uNtttt,uNttt)L2(Qτ)+α(uNttt,uNttt)L2(Qτ)+β(uNtt,uNttt)L2(Qτ)+ϱ(uN,uNttt)L2(Qτ)+δ(uNt,uNttt)L2(Qτ)+γ(uNtt,uNttt)L2(Qτ)(wN,uNttt)L2(Qτ)=(F,uNttt)L2(Qτ)+ϱτ0ΩuNttt(x,t)(t0ΩuN(ξ,η)dξdη)dsxdt+δτ0ΩuNttt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt+γτ0ΩuNttt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdtτ0ΩuNttt(x,t)(t0ΩwN(ξ,η)dξdη)dsxdt. (3.23)

    With the same reasoning in (3.9), we find

    (uNtttt,uNttt)L2(Qτ)=12uNτττ(x,τ)2L2(Ω)12uNttt(x,0)2L2(Ω)α(uNttt,uNttt)L2(Qτ)=ατ0uNttt(x,t)2L2(Ω),β(uNtt,uNttt)L2(Qτ)=β2uNττ(x,τ)2L2(Ω)β2uNtt(x,0)2L2(Ω),ϱ(uN,uNttt)L2(Qτ)=ϱ(uN(x,τ),uNττ(x,τ))L2(Ω)ϱ(uN(x,0),uNtt(x,0))L2(Ω)ϱτ0(uNt,uNtt)L2(Ω)dt,δ(uNt,uNttt)L2(Qτ)=δτ0uNtt(x,t)2L2(Ω)dt+δ(uNτ(x,τ),uNττ(x,τ))L2(Ω)δ(uNt(x,0),uNtt(x,0))L2(Ω),γ(uNtt,uNttt)L2(Qτ)=γ2uNττ(x,τ)2L2(Ω)γ2uNtt(x,0)2L2(Ω)(wN,uNttt)L2(Qτ)=H(τ)(uNττ(x,τ),uN(x,τ))2L2(Ω)+h(τ)(uNτ(x,τ),uN(x,τ))2L2(Ω)12uN(x,τ)2L2(Ω)+ΩuNτττ0h(τσ)(uN(τ)uN(σ))dσdx+ΩuNττ0h(τσ)(uN(τ)uN(σ))dσdx+12huN(τ)+12τ0(hh)uN(t)dth(0)τ0uNt(x,t)2L2(Ω)dt, (3.24)
    ϱτ0ΩuNttt(t0ΩuN(ξ,η)dξdη)dsxdt (3.25)
    =ϱΩuNττ(x,τ)τ0ΩuN(ξ,t)dξdtdsxϱΩτ0uNtt(x,t)ΩuN(ξ,t)dξdtdsx, (3.26)
    δτ0ΩuNttt(x,t)(t0ΩuNt(ξ,η)dξdη)dsxdt=δΩuNττ(x,τ)ΩuN(ξ,τ)dξdsxδΩuNττ(x,τ)ΩuN(ξ,0)dξdsxδΩτ0uNtt(x,t)ΩuNt(ξ,t)dξdtds, (3.27)
    γτ0ΩuNttt(x,t)(t0ΩuNtt(ξ,η)dξdη)dsxdt=γΩuNττ(x,τ)ΩuNτ(ξ,τ)dξdsxγΩuNττ(x,τ)ΩuNt(ξ,0)dξdsxγΩτ0uNtt(x,t)ΩuNtt(ξ,t)dξdtds, (3.28)
    τ0ΩuNttt(t0ΩwN(ξ,η)dξdη)dsxdt=τ0ΩuNttt(t0ΩH(η)uN(ξ,η)dξdη)dsxdt+τ0ΩuNttt(t0Ω[η0h(ησ)(uN(ξ,η)uN(ξ,σ))dσ]dξdη)dsxdt=ΩuNττ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+τ0ΩuNtt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuNττ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuNtt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.29)

    A substitution of equalities (3.24)-(3.29) in (3.23), gives

    12uNτττ(x,τ)2L2(Ω)+β2uNττ(x,τ)2L2(Ω)+ϱ(uN(x,τ),uNττ(x,τ))L2(Ω)+δ(uNτ(x,τ),uNττ(x,τ))L2(Ω)+γ2uNττ(x,τ)2L2(Ω)H(τ)(uNττ(x,τ),uN(x,τ))2L2(Ω)+h(τ)(uNτ(x,τ),uN(x,τ))2L2(Ω)12uN(x,τ)2L2(Ω)+ΩuNτττ0h(τσ)(uN(τ)uN(σ))dσdx+ΩuNττ0h(τσ)(uN(τ)uN(σ))dσdx+12huN(τ)=(F,uNttt)L2(Qτ)+12uNttt(x,0)2L2(Ω)ατ0uNttt(x,t)2L2(Ω)+ϱ(uN(x,0),uNtt(x,0))L2(Ω)+ϱτ0(uNt,uNtt)L2(Ω)dt+δτ0uNtt(x,t)2L2(Ω)dt+δ(uNt(x,0),uNtt(x,0))L2(Ω)+ϱΩuNττ(x,τ)τ0ΩuN(ξ,t)dξdtdsxγ2uNtt(x,0)2L2(Ω)ϱΩτ0uNtt(x,t)ΩuN(ξ,t)dξdtdsxβ2uNtt(x,0)2L2(Ω)+δΩuNττ(x,τ)ΩuN(ξ,τ)dξdsxδΩuNττ(x,τ)ΩuN(ξ,0)dξdsxδΩτ0uNtt(x,t)ΩuNt(ξ,t)dξdtds+δΩuNττ(x,τ)ΩuNτ(ξ,τ)dξdsxγΩuNττ(x,τ)ΩuNt(ξ,0)dξdsxγΩτ0uNtt(x,t)ΩuNtt(ξ,t)dξdtdsΩuNττ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+12τ0(hh)uN(t)dth(0)τ0uNt(x,t)2L2(Ω)dt+τ0ΩuNtt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+ΩuNττ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtτ0ΩuNtt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdt. (3.30)

    Multiplying (3.15) by λ1, (3.22) by λ2, and (3.30) by λ3 such as (λ1+λ2<λ3), we get

    λ1(uNτττ(x,τ),uNτ(x,τ))L2(Ω)+λ1α(uNττ(x,τ),uNτ(x,τ))L2(Ω)+λ1β2uNτ(x,τ)2L2(Ω)+λ1ϱ2uN(x,τ)2L2(Ω)+(λ1γ2+λ2δ2)uNτ(x,τ)2L2(Ω)+λ2(uNτττ(x,τ),uNττ(x,τ))L2(Ω)+(λ2α2+λ3β2)uNττ(x,τ)2L2(Ω)+λ2ϱ(uN(x,τ),uNτ(x,τ))L2(Ω)+λ32uNτττ(x,τ)2L2(Ω)+λ3ϱ(uN(x,τ),uNττ(x,τ))L2(Ω)+λ3δ(uNτ(x,τ),uNττ(x,τ))L2(Ω)+λ3γ2uNττ(x,τ)2L2(Ω)+λ12huN(τ)λ12H(τ)uN(x,τ)2L2(Ω)λ22huN(τ)+λ2(wN(τ),uNτ)L2(Ω)λ3H(τ)(uNττ(x,τ),uN(x,τ))2L2(Ω)+λ3h(τ)(uNτ(x,τ),uN(x,τ))2L2(Ω)λ32uN(x,τ)2L2(Ω)+λ3ΩuNτττ0h(τσ)(uN(τ)uN(σ))dσdx+λ3ΩuNττ0h(τσ)(uN(τ)uN(σ))dσdx+λ32huN(τ)=λ1(uNttt(x,0),uNt(x,0))L2(Ω)+λ1α(uNtt(x,0),uNt(x,0))L2(Ω)+λ1ϱ2uN(x,0)2L2(Ω)+λ1β2uNt(x,0)2L2(Ω)+(λ1γ2+λ2δ2)uNt(x,0)2L2(Ω)+λ1τ0(uNttt,uNtt)L2(Ω)dt+(λ1αλ2β)τ0utt(x,t)2L2(Ω)dt+(λ2ϱλ1δ)τ0uNt(x,t)2L2(Ω)dt+(λ2λ3α)τ0uNttt(x,t)2L2(Ω)dt+λ2(uNttt(x,0),uNtt(x,0))L2(Ω)+(λ2α2λ3β2)uNtt(x,0)2L2(Ω)λ32uNttt(x,0)2L2(Ω)+λ2ϱ(uN(x,0),uNt(x,0))L2(Ω)+(λ3δλ2γ)τ0uNtt(x,t)2L2(Ω)dt+λ3ϱ(uN(x,0),uNtt(x,0))L2(Ω)+λ3ϱτ0(uNt,uNtt)L2(Ω)dt+λ3δ(uNt(x,0),uNtt(x,0))L2(Ω)λ3γ2uNtt(x,0)2L2(Ω)+λ1ϱΩuN(x,τ)τ0ΩuN(ξ,t)dξdtdsxλ1ϱΩτ0uN(x,t)ΩuN(ξ,t)dξdtdsx+(λ1δλ2ϱ)Ωτ0uNt(x,t)ΩuN(ξ,t)dξdtdsxλ1δΩτ0uNt(x,t)ΩuN(ξ,0)dξdtdsx+(λ1γλ2δ)τ0ΩuNt(x,t)(ΩuNt(ξ,t)dξ)dsxdtλ1γτ0ΩuNt(x,t)(ΩuNt(ξ,0)dξ)dsxdt+λ2ϱΩuNτ(x,τ)τ0ΩuN(ξ,t)dξdtdsx+λ2δΩuNτ(x,τ)ΩuN(ξ,τ)dξdsxλ2δΩuNτ(x,τ)ΩuN(ξ,0)dξdsx+λ2γΩuNτ(x,τ)ΩuNτ(ξ,τ)dξdsxλ2γΩuNτ(x,τ)ΩuNt(ξ,0)dξdsxλ2γΩτ0uNt(x,t)ΩuNtt(ξ,t)dξdtdsx+λ3ϱΩuNττ(x,τ)τ0ΩuN(ξ,t)dξdtdsxλ3ϱΩτ0uNtt(x,t)ΩuN(ξ,t)dξdtdsx+λ3δΩuNττ(x,τ)ΩuN(ξ,τ)dξdsxλ3δΩuNττ(x,τ)ΩuN(ξ,0)dξdsxλ3δΩτ0uNtt(x,t)ΩuNt(ξ,t)dξdtds+λ3γΩuNττ(x,τ)ΩuNτ(ξ,τ)dξdsxλ3γΩuNττ(x,τ)ΩuNt(ξ,0)dξdsxλ3γΩτ0uNtt(x,t)ΩuNtt(ξ,t)dξdtds+λ1(F,uNt)L2(Qτ)+λ2(F,uNtt)L2(Qτ)+λ3(F,uNttt)L2(Qτ)+λ12τ0huN(t)dtλ12h(t)uN(x,t)2L2(Ω)dtλ1ΩuN(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+λ1τ0ΩuN(x,t)ΩH(t)uN(ξ,t)dξdsxdt+λ1ΩuN(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtλ1τ0ΩuN(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdtλ3ΩuNττ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsxλ32τ0(hh)uN(t)dt+λ3h(0)τ0uNt(x,t)2L2(Ω)dt+λ3τ0ΩuNtt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+λ3ΩuNττ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtλ3τ0ΩuNtt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdtλ2ΩuNτ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsx+λ2τ0ΩuNt(x,t)ΩH(t)uN(ξ,t)dξdsxdt+λ2ΩuNτ(x,τ)(τ0Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξ)dsxdtλ2τ0ΩuNt(x,t)Ω[t0h(tσ)(uN(ξ,t)uN(ξ,σ))dσ]dξdsxdtλ22τ0huN(t)dt+λ22τ0h(t)uN(x,t)2L2(Ω)dt. (3.31)

    We can estimate all the terms in the RHS of (3.31) as follows

    λ1ϱΩuN(x,τ)τ0ΩuN(ξ,t)dξdtdsxλ1ϱ2ε1(εuN(x,τ)2L2(Ω)+l(ε)uN(x,τ)2L2(Ω))+λ1ϱ2ε1T|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.32)
    λ1ϱΩτ0uN(x,t)ΩuN(ξ,t)dξdtdsxλ1ϱ2ετ0uN(x,t)2L2(Ω)dt+λ1ϱ2(l(ε)+|Ω||Ω|)τ0uN(x,t)2L2(Ω)dt, (3.33)
    (λ1δλ2ϱ)Ωτ0uNt(x,t)ΩuN(ξ,t)dξdtdsx(λ1δ+λ2ϱ)2(ετ0uNt(x,t)2L2(Ω)dt+l(ε)τ0uNt(x,t)2L2(Ω)dt)+(λ1δ+λ2ϱ)2|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.34)
    λ1δΩτ0uNt(x,t)ΩuN(ξ,0)dξdtdsxλ1δ2(ετ0uNt(x,t)2L2(Ω)dt+l(ε)τ0uNt(x,t)2L2(Ω)dt)+λ1δ2|Ω||Ω|TuN(x,0)2L2(Ω), (3.35)
    λ2ϱΩuNτ(x,τ)τ0ΩuN(ξ,t)dξdtdsxλ2ϱ2(εε2uNτ(x,τ)2L2(Ω)+l(ε)ε2uNτ(x,τ)2L2(Ω))+λ2ϱ2ε2|Ω||Ω|Tτ0uN(x,t)2L2(Ω)dt, (3.36)
    λ2δΩuNτ(x,τ)ΩuN(ξ,τ)dξdsxλ2δ2ε3(εuNτ(x,τ)2L2(Ω)+l(ε)uNτ(x,τ)2L2(Ω))+λ2δ2ε3|Ω||Ω|uN(x,τ)2L2(Ω), (3.37)
    λ2δΩuNτ(x,τ)ΩuN(ξ,0)dξdsxλ2δ2ε4(εuNτ(x,τ)2L2(Ω)+l(ε)uNτ(x,τ)2L2(Ω))+λ2δ2ε4|Ω||Ω|uN(x,0)2L2(Ω), (3.38)
    (λ1γλ2δ)Ωτ0uNt(x,t)ΩuNt(ξ,t)dξdtdsx(λ1γ+λ2δ)2ετ0uNt(x,t)2L2(Ω)dt+(λ1γ+λ2δ)2(l(ε)+|Ω||Ω|)τ0uNt(x,t)2L2(Ω)dt, (3.39)
    λ1γτ0ΩuNt(x,t)(ΩuNt(ξ,0)dξ)dsxdtλ1γ2(ετ0uNt(x,t)2L2(Ω)dt+l(ε)τ0uNt(x,t)2L2(Ω)dt)+λ1γ2|Ω||Ω|TuNt(x,0)2L2(Ω), (3.40)
    λ2γΩuNτ(x,τ)ΩuNτ(ξ,τ)dξdsxλ2γ2ε5(εuNτ(x,τ)2L2(Ω)+l(ε)uNτ(x,τ)2L2(Ω))+λ2γ2ε5|Ω||Ω|uNτ(x,τ)2L2(Ω), (3.41)
    λ2γΩuNτ(x,τ)ΩuNt(ξ,0)dξdsxλ2γ2ε6(εuNτ(x,τ)2L2(Ω)+l(ε)uNτ(x,τ)2L2(Ω))+λ2γ2ε6|Ω||Ω|uNt(x,0)2L2(Ω), (3.42)
    λ2γΩτ0uNt(x,t)ΩuNtt(ξ,t)dξdtdsxλ2γ2ετ0uNt(x,t)2L2(Ω)dt+λ2γ2l(ε)τ0uNt(x,t)2L2(Ω)dt+λ2γ2|Ω||Ω|τ0uNtt(x,t)2L2(Ω)dt, (3.43)
    λ3ϱΩuNττ(x,τ)τ0ΩuN(ξ,t)dξdtdsxλ3ϱ2(εε7uNττ(x,τ)2L2(Ω)+l(ε)ε7uNττ(x,τ)2L2(Ω))+λ3ϱ2ε7|Ω||Ω|Tτ0uN(x,t)2L2(Ω)dt, (3.44)

    and

    λ3ϱΩτ0uNtt(x,t)ΩuN(ξ,t)dξdtdsxλ3ϱ2(ετ0uNtt(x,t)2L2(Ω)dt+l(ε)τ0uNtt(x,t)2L2(Ω)dt)+λ3ϱ2|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.45)
    λ3δΩuNττ(x,τ)ΩuN(ξ,τ)dξdsxλ3δ2ε8(εuNττ(x,τ)2L2(Ω)+l(ε)uNττ(x,τ)2L2(Ω))+λ3δ2ε8|Ω||Ω|uN(x,τ)2L2(Ω), (3.46)
    λ3δΩuNττ(x,τ)ΩuN(ξ,0)dξdsxλ3δ2ε9(εuNττ(x,τ)2L2(Ω)+l(ε)uNττ(x,τ)2L2(Ω))+λ3δ2ε9|Ω||Ω|uN(x,0)2L2(Ω), (3.47)
    λ3δΩτ0uNtt(x,t)ΩuNt(ξ,t)dξdtdsxλ3δ2(ετ0uNtt(x,t)2L2(Ω)dt+l(ε)τ0uNtt(x,t)2L2(Ω)dt)+λ3δ2|Ω||Ω|τ0uNt(x,t)2L2(Ω)dt, (3.48)
    λ3γΩuNττ(x,τ)ΩuNτ(ξ,τ)dξdsxλ3γ2ε10(εuNττ(x,τ)2L2(Ω)+l(ε)uNττ(x,τ)2L2(Ω))+λ3γ2ε10|Ω||Ω|uNτ(x,τ)2L2(Ω), (3.49)
    λ3γΩuNττ(x,τ)ΩuNt(ξ,0)dξdsxλ3γ2ε11(εuNττ(x,τ)2L2(Ω)+l(ε)uNττ(x,τ)2L2(Ω))+λ3γ2ε11|Ω||Ω|uNt(x,0)2L2(Ω), (3.50)
    λ3γΩτ0uNtt(x,t)ΩuNtt(ξ,t)dξdtdsxλ3γ2ετ0uNtt(x,t)2L2(Ω)dt+λ3γ2(l(ε)+|Ω||Ω|)τ0uNtt(x,t)2L2(Ω)dt, (3.51)
    λ12uNτττ(x,τ)2L2(Ω)λ12uNτ(x,τ)2L2(Ω)λ1(uNτττ(x,τ),uNτ(x,τ))L2(Ω), (3.52)
    λ22uNτττ(x,τ)2L2(Ω)λ22uNττ(x,τ)2L2(Ω)λ2(uNτττ(x,τ),uNττ(x,τ))L2(Ω), (3.53)
    λ1α2uNττ(x,τ)2L2(Ω)λ1α2uNτ(x,τ)2L2(Ω)λ1α(uNττ(x,τ),uNτ(x,τ))L2(Ω), (3.54)
    λ2ϱε122uN(x,τ)2L2(Ω)λ2ϱ2ε12uNτ(x,τ)2L2(Ω)λ2ϱ(uN(x,τ),uNτ(x,τ))L2(Ω), (3.55)
    λ2ϱε132uN(x,τ)2L2(Ω)λ2ϱ2ε13uNττ(x,τ)2L2(Ω)λ3ϱ(uN(x,τ),uNττ(x,τ))L2(Ω), (3.56)
    λ3δε142uNτ(x,τ)2L2(Ω)λ3δ2ε14uNττ(x,τ)2L2(Ω)λ3δ(uNτ(x,τ),uNττ(x,τ))L2(Ω), (3.57)
    λ1(uNttt(x,0),uNt(x,0))L2(Ω)λ12uNttt(x,0)2L2(Ω)+λ12uNt(x,0)2L2(Ω) (3.58)
    λ1α(uNtt(x,0),uNt(x,0))L2(Ω)λ1α2uNtt(x,0)2L2(Ω)+λ1α2uNt(x,0)2L2(Ω), (3.59)
    λ2(uNttt(x,0),uNtt(x,0))L2(Ω)λ22uNttt(x,0)2L2(Ω)+λ22uNtt(x,0)2L2(Ω), (3.60)
    λ2ϱ(uN(x,0),uNt(x,0))L2(Ω)λ22ϱuN(x,0)2L2(Ω)+λ22ϱuNt(x,0)2L2(Ω), (3.61)
    λ3ϱ(uN(x,0),uNtt(x,0))L2(Ω)λ32ϱuN(x,0)2L2(Ω)+λ32ϱuNtt(x,0)2L2(Ω), (3.62)
    λ3δ(uNt(x,0),uNtt(x,0))L2(Ω)λ32δuNt(x,0)2L2(Ω)+λ32δuNtt(x,0)2L2(Ω), (3.63)
    λ1τ0(uNttt,uNtt)L2(Ω)dtλ12τ0uNttt(x,t)2L2(Ω)dt+λ12τ0uNtt(x,t)2L2(Ω)dt, (3.64)
    λ3ϱτ0(uNt,uNtt)L2(Ω)dtλ3ϱ2τ0uNt(x,t)2L2(Ω)dt+λ3ϱ2τ0uNtt(x,t)2L2(Ω)dt, (3.65)
    λ1ΩuN(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsxλ1h02(εuN(x,τ)2L2(Ω)+l(ε)uN(x,τ)2L2(Ω))+λ1h02T|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.66)
    λ1Ωτ0uN(x,t)ΩH(t)uN(ξ,t)dξdtdsxλ1h02ετ0uN(x,t)2L2(Ω)dt+λ1h02(l(ε)+|Ω||Ω|)τ0uN(x,t)2L2(Ω)dt, (3.67)
    λ1ΩuN(x,τ)τ0Ωt0h(tσ)(uN(ξ,t)uN(ξ,σ))dσdξdtdsxλ1h02(εuN(x,τ)2L2(Ω)+l(ε)uN(x,τ)2L2(Ω))+λ12T|Ω||Ω|τ0huN(t)dt, (3.68)
    λ1Ωτ0uN(x,t)Ωt0h(tσ)(uN(ξ,t)uN(ξ,σ))dσdξdtdsxλ1h02ετ0uN(x,t)2L2(Ω)dt+λ1h02l(ε)τ0uN(x,t)2L2(Ω)dt+λ12|Ω||Ω|τ0huN(t)dt, (3.69)
    λ2ΩuNτ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsxλ2h02(εuNτ(x,τ)2L2(Ω)+l(ε)uNτ(x,τ)2L2(Ω))+λ2h02T|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.70)
    λ2Ωτ0uNτ(x,t)ΩH(t)uN(ξ,t)dξdtdsxλ2h02ετ0uNt(x,t)2L2(Ω)dt+λ2h02l(ε)τ0uNt(x,t)2L2(Ω)dt+λ2h02|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.71)
    λ2ΩuNτ(x,τ)τ0Ωt0h(tσ)(uN(ξ,t)uN(ξ,σ))dσdξdtdsxλ2h02(εuNτ(x,τ)2L2(Ω)+l(ε)uNτ(x,τ)2L2(Ω))+λ22T|Ω||Ω|τ0huN(t)dt, (3.72)
    λ2Ωτ0uNt(x,t)Ωt0h(tσ)(uN(ξ,t)uN(ξ,σ))dσdξdtdsxλ2h02ετ0uNt(x,t)2L2(Ω)dt+λ22h0l(ε)τ0uNt(x,t)2L2(Ω)dt+λ22|Ω||Ω|τ0huN(t)dt, (3.73)
    λ3ΩuNττ(x,τ)τ0ΩH(t)uN(ξ,t)dξdtdsxλ3h02ε18(εuNττ(x,τ)2L2(Ω)+l(ε)uNττ(x,τ)2L2(Ω))+λ3h02ε18T|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.74)
    λ3Ωτ0uNττ(x,t)ΩH(t)uN(ξ,t)dξdtdsxλ3h02ετ0uNtt(x,t)2L2(Ω)dt+λ3h02l(ε)τ0uNtt(x,t)2L2(Ω)dt+λ3h02|Ω||Ω|τ0uN(x,t)2L2(Ω)dt, (3.75)
    λ3ΩuNττ(x,τ)τ0Ωt0h(tσ)(uN(ξ,t)uN(ξ,σ))dσdξdtdsxλ3h02ε15(εuNττ(x,τ)2L2(Ω)+l(ε)uNττ(x,τ)2L2(Ω))+λ32ε15T|Ω||Ω|τ0huN(t)dt, (3.76)
    λ3Ωτ0uNtt(x,t)Ωt0h(tσ)(uN(ξ,t)uN(ξ,σ))dσdξdtdsxλ3h02ετ0uNtt(x,t)2L2(Ω)dt+λ32h0l(ε)τ0uNtt(x,t)2L2(Ω)dt+λ32|Ω||Ω|τ0huN(t)dt, (3.77)
    λ3H(τ)(uNττ,uN)2L2(Ω)λ3h02ε16uNττ(x,τ)2L2(Ω)λ3h0ε162uN(x,τ)2L2(Ω), (3.78)
    λ3h(τ)(uNτ,uN)2L2(Ω)λ3h(0)2uNτ(x,τ)2L2(Ω)λ3h(0)2uN(x,τ)2L2(Ω), (3.79)
    λ3ΩuNττ[τ0h(τσ)(uN(τ)uN(σ)dσ]dxλ3h02ε17uNττ(x,τ)2L2(Ω)λ3ε172huN(τ), (3.80)
    λ3ΩuNτ[τ0h(τσ)(uN(τ)uN(σ)dσ]dxλ3h02uNτ(x,τ)2L2(Ω)+λ32huN(τ), (3.81)
    λ2ΩuNτ[τ0h(τσ)uN(σ)dσ]dxλ22huN(τ)λ2(h0+1)2uNτ(x,τ)2L2(Ω)λ2h02uN(x,τ)2L2(Ω), (3.82)
    λ1(F,uNt)L2(Qτ)λ12τ0F(x,t)2L2(Ω)dt+λ12τ0ut(x,t)2L2(Ω)dtλ2(F,uNtt)L2(Qτ)λ22τ0F(x,t)2L2(Ω)dt+λ22τ0utt(x,t)2L2(Ω)dtλ3(F,uNttt)L2(Qτ)λ32τ0F(x,t)2L2(Ω)dt+λ32τ0uttt(x,t)2L2(Ω)dt. (3.83)

    Substituting (3.32)-(3.83) into (3.31) and make use of the following inequality

    m1uN(x,τ)2L2(Ω)m1uN(x,t)2L2(Qτ)+m1uNt(x,t)2L2(Qτ)+m1uN(x,0)2L2(Ω)m2uNτ(x,τ)2L2(Ω)m2uNt(x,t)2L2(Qτ)+m2uNtt(x,t)2L2(Qτ)+m2uNt(x,0)2L2(Ω)m3uNττ(x,τ)2L2(Ω)m3uNtt(x,t)2L2(Qτ)+m3uNttt(x,t)2L2(Qτ)+m3uNtt(x,0)2L2(Ω)m4uN(x,τ)2L2(Ω)m4uN(x,t)2L2(Qτ)+m4uNt(x,t)2L2(Qτ)+m4uN(x,0)2L2(Ω)m5uNτ(x,τ)2L2(Ω)m5uNt(x,t)2L2(Qτ)+m5uNtt(x,t)2L2(Qτ)+m5uNt(x,0)2L2(Ω)m6huN(τ)m6uNt(x,t)2L2(Qτ)+m6τ0huN(t)dtm7huN(τ)m7uNt(x,t)2L2(Qτ)+m7τ0huN(t)dtm8huN(τ)m8uNt(x,t)2L2(Qτ)m8τ0huN(t)dt,

    where

    m1=λ1ϱε1l(ε)+λ2δ2ε3|Ω||Ω|+λ3δ2ε8|Ω||Ω|+λ1h0l(ε),m2=λ2ϱ2l(ε)ε2+λ2δ2l(ε)ε3+λ2δ2l(ε)ε4+λ2γ2(l(ε)ε5+ε5|Ω||Ω|)+λ2γ2l(ε)ε6+λ3γ2ε10|Ω||Ω|+λ1(1+α)2+λ2h0l(ε),m3=λ3ϱ2l(ε)ε7+λ3δ2l(ε)ε8+λ3δ2l(ε)ε9+λ3γ2l(ε)ε10+λ3γ2l(ε)ε11+λ22+λ1α2+λ3h02ε18l(ε)+λ32ε15l(ε),m4=λ1h02ε1ε+λ2ϱ2ε12+λ2ϱ2ε13+λ1h0ε+λ32+λ3h02ε16+λ3h(0)2+λ2h02+λ1ϱ2ε1ε,m5=λ2ϱ2εε2+λ2δ2εε3+λ2δ2εε4+λ2γ2εε5+λ2γ2εε6+λ2ϱ2ε12+λ3δε142+λ2h0ε+λ3(h0+h(0))2+λ2(h0+1)2,m7=λ2ε172+λ22,m8=λ32,m6=1,

    we have

    λ1ϱ2ε1l(ε)uN(x,τ)2L2(Ω)+λ1β2uNτ(x,τ)2L2(Ω)+(λ2α2+λ3β2)uNττ(x,τ)2L2(Ω)+{λ32λ12λ22}uNτττ(x,τ)2L2(Ω)+λ1ϱ2uN(x,τ)2L2(Ω)+{λ1γ2+λ2δ2}uNτ(x,τ)2L2(Ω)+huN(τ)+λ12huN(τ)λ22huN(τ)+{λ3γ2λ3ϱ2εε7λ3δ2εε8λ3δ2εε9λ3γ2εε10λ3γ2εε11λ2ϱ2ε13λ3δ2ε14λ3h02εε16λ3h02εε17λ3h02ε18λ3h02ε15}uNττ(x,τ)2L2(Ω)γ7uN(x,0)2L2(Ω)+{λ22+λ1α2+(λ2α2λ3β2)+m3}uNtt(x,0)2L2(Ω)+{λ12+λ22+λ32}uNttt(x,0)2L2(Ω)+{λ1ϱ2+λ2ϱ2+λ3ϱ2+m4}uN(x,0)2L2(Ω)+γ8uNt(x,0)2L2(Ω)+{λ2ϱ2+λ3δ2+λ1γ2+λ2δ2+m5}uNt(x,0)2L2(Ω)+{λ3ϱ2+3λ3δ2λ3γ2λ2γ}uNtt(x,0)2L2(Ω)+(γ1+m1)τ0uN(x,t)2L2(Ω)dt+(γ2+m1+m2)τ0uNt(x,t)2L2(Ω)dt+{λ12+λ2λ3α+m3}τ0uNttt(x,t)2L2(Ω)dtm8τ0huN(t)dt+{γ6+m4}τ0uN(x,t)2L2(Ω)dt+(γ3+m2+m3)τ0uNtt(x,t)2L2(Ω)dt+(γ4+m4+m5+m7+m8)τ0uNt(x,t)2L2(Ω)dt+(γ5+m5)τ0uNtt(x,t)2L2(Ω)dt+τ0huN(t)dt+m7τ0huN(t)dt+λ1+λ2+λ32τ0F(x,t)2L2(Ω)dt, (3.84)

    where

    γ1=λ1ϱ2ε1T|Ω||Ω|+λ1ϱ2(l(ε)+|Ω||Ω|)+(λ1δ+λ2ϱ2)|Ω||Ω|+λ2ϱ2ε2T|Ω||Ω|+λ3ϱ2ε7T|Ω||Ω|+λ3ϱ2|Ω||Ω|++λ1h02l(ε)+[λ3h02ε18+(λ3+λ2+λ1)h02+(λ1+λ2)h0T2]|Ω||Ω|γ2=(λ1δ+λ2ϱ2)l(ε)+λ1δ2l(ε)+(λ1γ+λ2δ2)(l(ε)+|Ω||Ω|)+λ1γ2l(ε)+λ2γ2l(ε)+λ3δ2|Ω||Ω|+λ2h0l(ε),
    γ3=λ2γ2|Ω||Ω|+λ3ϱ2l(ε)+λ3δ2l(ε)+λ3γ2(l(ε)+|Ω||Ω|)+λ12+(λ1αλ2β)+λ3ϱ2+λ3h0l(ε),γ4=(λ1δ+λ2ϱ2)ε+λ1δ2ε+(λ1γ+λ2δ2)ε+λ1γ2ε+λ2γ2ε+λ3ϱ2+(λ2ϱλ1δ)+h(0)λ3+λ3h0ε,γ5=λ3δ2ε+λ3γ2ε+λ3ϱ2+(λ3δλ2γ)+λ3h0ε,γ6=λ1ϱ2ε+λ1h0ε,γ7=λ1δ2|Ω||Ω|T+λ2δ2ε4|Ω||Ω|+λ3δ2ε9|Ω||Ω|+m1,γ8=λ1γ2|Ω||Ω|T+λ2γ2ε6|Ω||Ω|+λ3γ2ε11|Ω||Ω|+λ12+λ1α2+λ1β2+m2.

    Choosing ε7, ε8, ε9, ε10, ε11, ε13, ε14,ε15, ε16, ε17 and ε18 sufficiently large

    β0:=λ3γ2λ3ϱ2εε7λ3δ2εε8λ3δ2εε9λ3γ2εε10λ3γ2εε11λ3δ2ε14λ2ϱ2ε13λ3h02εε16λ32εε17λ3h02ε18λ32ε15>0, (3.85)

    the relation (3.84) reduces to

    {uN(x,τ)2L2(Ω)+uN(x,τ)2L2(Ω)+uNτ(x,τ)2L2(Ω)+uNτ(x,τ)2L2(Ω)+uNττ(x,τ)2L2(Ω)+uNττ(x,τ)2L2(Ω)+uNτττ(x,τ)2L2(Ω)+huN(τ)+huN(τ)huN(τ)} (3.86)
    Dτ0{uN(x,t)2L2(Ω)+uN(x,t)2L2(Ω)+uNt(x,t)2L2(Ω)+uNt(x,t)2L2(Ω)+uNtt(x,t)2L2(Ω)+uNtt(x,t)2L2(Ω)+uNttt(x,t)2L2(Ω)+huN(t)+huN(t)huN(t)+F2L2(Ω)}dt+D{uN(x,0)2W12(Ω)+uNt(x,0)2W12(Ω)+uNtt(x,0)2W12(Ω)+uNttt(x,0)2L2(Ω)+huN(0)+huN(0)huN(0)}, (3.87)

    where

    D:=max{λ1δ2|Ω||Ω|T+λ2δ2ε4|Ω||Ω|+λ3δ2ε9|Ω||Ω|+m1,λ1γ2|Ω||Ω|T+λ2γ2ε6|Ω||Ω|+λ3γ2ε11|Ω||Ω|+λ12+λ1α2+λ1β2+m2,λ22+λ1α2+λ2α2λ3β2+m3,λ1+λ2+λ32,λ1ϱ2+λ2ϱ2+λ3ϱ2+m4,λ2ϱ2+λ3δ2+λ1γ2+λ2δ2+m5,γ1+m1,γ2+m1+m2,γ3+m2+m3,λ12+λ2λ3α+m3,λ3ϱ2+λ3δ2λ3γ2,γ6+m4,γ4+m4+m5,γ5+m5,m7,m8,1}min{λ1ϱ2ε1l(ε),λ1β2,λ2α2+λ3β2,λ32λ12λ22,λ1ϱ2,λ1γ2+λ2δ2,1,λ12,λ22,β0}. (3.88)

    Applying the Gronwall inequality to (3.87) and then integrate from 0 to τ appears that

    uN(x,t)2W12(Qτ)+uNt(x,t)2W12(Qτ)+uNtt(x,t)2W12(Qτ)+uN(x,t)hDeDT{u0(x)2W12(Ω)+u1(x)2W12(Ω)+u2(x)2L2(Ω)+u3(x)2L2(Ω)+F2L2(Ω)}.. (3.89)

    We deduce from (3.89) that

    uN(x,t)2W12(Qτ)+uNt(x,t)2W12(Qτ)+uNtt(x,t)2W12(Qτ)+uN(x,t)hA, (3.90)

    where

    uN(x,t)h:=τ0(huN(t)+huN(t)huN(t))dt.

    Therefore the sequence {uN}N1 is bounded in V(QT), and we can extract from it a subsequence for which we use the same notation which converges weakly in V(QT)  to a limit function u(x,t) we have to show that u(x,t) is a generalized solution of (1.1). Since uN(x,t)u(x,t) in L2(QT) and uN(x,0)ζ(x) in L2(Ω), then u(x,0)=ζ(x).

    Now to prove that (2.1) holds, we multiply each of the relations (3.5) by a function pl(t)W12(0,T), pl(t)=0, then add up the obtained equalities ranging from l=1 to l=N, and integrate over t on (0,T). If we let ηN=Nk=1pk(t)Zk(x), then we have

    (uNttt,ηNt)L2(QT)α(uNtt,ηNt)L2(QT)β(uNt,ηNt)L2(QT)+ϱ(uN,ηN)L2(QT)+δ(uNt,ηN)L2(QT)γ(uNt,ηNt)L2(QT)(wN,ηN)L2(QT)=ϱΩT0ηN(x,t)(t0ΩuN(ξ,τ)dξdτ)dtdsx+δΩT0ηN(x,t)ΩuN(ξ,t)dξdtdsxδΩT0ηN(x,t)ΩuN(ξ,0)dξdtdsxγT0ΩηNt(ΩuN(ξ,t)dξ)dsxdt+γT0ΩηNt(ΩuN(ξ,0)dξ)dsxdtγ(ΔuNt(x,0),ηN(0))L2(Ω)ΩT0ηN(x,t)(t0ΩwN(ξ,τ)dξdτ)dtdsx+(F,ηNt)L2(QT)+(uNttt(x,0),ηN(0))L2(Ω)+α(uNtt(x,0),ηN(0))L2(Ω)+β(uNtt(x,0),ηN(0))L2(Ω), (3.91)

    for all ηN of the form Nk=1pl(t)Zk(x).

    Since

    t0Ω((uN(ξ,τ)u(ξ,τ))dξdτT|Ω|uNuL2(QT),
    T0ηN(x,t)Ω(uNt(ξ,t)ut(ξ,t))dξdt|Ω|(T0(ηN(x,t))2dt)1/2uNtutL2(QT),
    T0ηN(x,t)Ω(u(N(ξ,0)u(ξ,0))dξdt|Ω|(T0(ηN(x,t))2dt)1/2uN(x,0)u(x,0)L2(QT),

    and

    uNuL2(QT)0, asN,

    therefore we have

    ϱΩT0ηN(x,t)t0ΩuN(ξ,τ)dξdτdtdsxϱΩT0η(x,t)t0Ωu(ξ,τ)dξdτdtdsx,
    δΩT0ηN(x,t)ΩuN(ξ,t)dξdtdsxδΩT0η(x,t)Ωu(ξ,t)dξdtdsx,
    δΩT0ηN(x,t)ΩuN(ξ,0)dξdtdsδΩT0η(x,t)Ωu(ξ,0)dξdtds,
    γT0ΩηNt(ΩuN(ξ,t)dξ)dsxdtγT0Ωηt(Ωu(ξ,t)dξ)dsxdt,
    γT0ΩηNt(ΩuN(ξ,0)dξ)dsxdtγT0Ωηt(Ωu(ξ,0)dξ)dsxdt.
    ΩT0ηN(x,t)t0ΩwN(ξ,τ)dξdτdtdsxϱΩT0η(x,t)t0Ωw(ξ,τ)dξdτdtdsx.

    Thus, the limit function u satisfies (2.1) for every ηN=Nk=1pl(t)Zk(x). We denote by QN the totality of all functions of the form ηN=Nk=1pl(t)Zk(x), with pl(t)W12(0,T), pl(t)=0.

    But Nl=1QN is dense in W(QT), then relation (2.1) holds for all u W(QT). Thus we have shown that the limit function u(x,t) is a generalized solution of problem (1.1) in V(QT).

    Theorem 2. The problem (1.1) cannot have more than one generalized solution in V(QT).

    Proof. Suppose that there exist two different generalized solutions u1V(QT) and u2V(QT) for the problem (1.1). Then, U=u1u2 solves

    {Utttt+αUttt+βUttϱΔUδΔUtγΔUtt+t0h(tσ)Δu(σ)dσ=0,U(x,0)=Ut(x,0)=Utt(x,0)=Uttt(x,0)=0uη=t0Ωu(ξ,τ)dξdτ,   xΩ. (4.1)

    and (2.1) gives

    (Uttt,vt)L2(QT)α(Utt,vt)L2(QT)β(Ut,vt)L2(QT)+ϱ(U,v)L2(QT)+δ(Ut,v)L2(QT)γ(Ut,vt)L2(QT)(W,v)L2(QT)=ϱT0Ωv(t0Ωu(ξ,τ)dξdτ)dsxdt+δT0ΩvΩU(ξ,t)dξdsxdtγT0Ωvt(ΩUτ(ξ,t)dξdt)dsxdtT0Ωv(t0ΩW(ξ,τ)dξdτ)dsxdt, (4.2)

    where

    W(x,t):=t0h(tσ)ΔU(σ)dσ.

    Consider the function

    v(x,t)={τtU(x,s)ds,0tτ,0,τtT. (4.3)

    It is obvious that vW(QT) and vt(x,t)=U(x,t) for all t[0,τ]. Integration by parts in the left hand side of (4.2) gives

    (Uttt,vt)L2(QT)=(Uττ(x,τ),U(x,τ))L2(Ω)12Uτ(x,τ)2L2(Ω), (4.4)
    α(Utt,vt)L2(QT)=α(Uτ(x,τ),U(x,τ))L2(Ω)ατ0Ut(x,t)2L2(Ω)dt, (4.5)
    β(Ut,vt)L2(QT)=β2U(x,τ)2L2(Ω), (4.6)
    ϱ(U,v)L2(QT)=ϱ2v(x,0)2L2(Ω), (4.7)
    δ(Ut,v)L2(QT)=δτ0vt(x,t)2L2(Ω)dt, (4.8)
    γ(Ut,vt)L2(QT)=γ2U(x,τ)2L2(Ω), (4.9)
    (W,v)L2(QT)h0τ0v(x,t)2L2(Ω)dt+h02τ0U(x,t)2L2(Ω)dt+12τ0hU(t)dt. (4.10)

    Plugging (4.4)-(4.10) into (4.2) we get

    (Uττ(x,τ),U(x,τ))L2(Ω)+α(Uτ(x,τ),U(x,τ))L2(Ω)+β2U(x,τ)2L2(Ω)+ϱ2v(x,0)2L2(Ω)+γ2U(x,τ)2L2(Ω)12Uτ(x,τ)2L2(Ω)ατ0Ut(x,t)2L2(Ω)dtδτ0vt(x,t)2L2(Ω)dt+h0τ0v(x,t)2L2(Ω)dt+h02τ0U(x,t)2L2(Ω)dt+12τ0hU(t)dt+ϱT0Ωv(t0ΩU(ξ,τ)dξdτ)dsxdt+δT0ΩvΩU(ξ,t)dξdsxdtγT0Ωvt(ΩU(ξ,t)dξ)dsdtT0Ωv(t0ΩW(ξ,τ)dξdτ)dsxdt. (4.11)

    Now since

    v2(x,t)=(τtU(x,s)ds)2ττ0U2(x,s)ds,

    then

    v2L2(Qτ)τ2U2L2(Qτ)T2U2L2(Qτ). (4.12)

    Using the trace inequality, the RHS of (4.11) can be estimated as follows

    ϱT0Ωv(t0ΩU(ξ,τ)dξdτ)dsxdtϱ2T2{l(ε)+|Ω||Ω|}τ0U(x,t)2L2(Ω)dt+ϱ2ετ0v(x,t)2L2(Ω)dt, (4.13)

    and

    δT0ΩvΩU(ξ,t)dξdsxdtδ2{T2l(ε)+|Ω||Ω|}τ0U(x,t)2L2(Ω)dt+δ2ετ0v(x,t)2L2(Ω)dt, (4.14)

    and

    γT0Ωvt(ΩU(ξ,t)dξ)dsdt=γτ0Ωv(ΩUt(ξ,t)dξ)dsdtγ|Ω||Ω|2Ut2L2(Qτ)+γT22εv2L2(Qτ)+γ2l(ε)T2U2L2(Qτ). (4.15)
    T0Ωv(t0ΩW(ξ,τ)dξdτ)dsxdt=T0Ωv(t0ΩH(τ)U(ξ,τ)dξdτ)dsxdt+T0Ωv(t0Ω[τ0h(τσ)(U(ξ,τ)U(ξ,σ))dσ]dξdτ)dsxdth02T2{l(ε)+|Ω||Ω|}τ0U(x,t)2L2(Ω)dt+h02ετ0v(x,t)2L2(Ω)dt+12l(ε)τ0U(x,t)2L2(Ω)dt+12|Ω||Ω|τ0hU(t)dt+12ετ0v(x,t)2L2(Ω)dt. (4.16)

    Combining the relations (4.13)-(4.16) and (4.11) we get

    (Uττ(x,τ),U(x,τ))L2(Ω)+α(Uτ(x,τ),U(x,τ))L2(Ω)+β2U(x,τ)2L2(Ω)+ϱ2v(x,0)2L2(Ω)+γ2U(x,τ)2L2(Ω)12Uτ(x,τ)2L2(Ω){ϱ2T2(l(ε)+|Ω||Ω|)+δ2(T2l(ε)+|Ω||Ω|)+γ2l(ε)T2+h02T2(l(ε)+|Ω||Ω|)+12l(ε)}τ0U(x,t)2L2(Ω)dt+(α+γ|Ω||Ω|2)τ0Ut(x,t)2L2(Ω)dt+12τ0Utt(x,t)2L2(Ω)dt+{(ϱ+δ+γ+h02)ε+h0}τ0v(x,t)2L2(Ω)dt+h0τ0v(x,t)2L2(Ω)dt+h02τ0U(x,t)2L2(Ω)dt+12τ0hU(t)dt+12|Ω||Ω|τ0hU(t)dt. (4.17)

    Next, multiplying the differential equation in (4.1) by Uttt and integrating over Qτ=Ω×(0,τ), we obtain

    (Utttt,Uttt)L2(Qτ)+α(Uttt,Uttt)L2(Qτ)+β(Utt,Uttt)L2(Qτ)ϱ(ΔU,Uttt)L2(Qτ)δ(ΔUt,Uttt)L2(Qτ)γ(ΔUt,Uttt)L2(Qτ)+(ΔW,Uttt)L2(Qτ)=0. (4.18)

    An integration by parts in (4.18) yields

    (Utttt,Uttt)L2(Qτ)=12Uτττ(x,τ)2L2(Ω), (4.19)
    α(Uttt,Uttt)L2(Qτ)=ατ0Uttt(x,t)2L2(Ω)dt, (4.20)
    β(Utt,Uttt)L2(Qτ)=β2Uττ(x,τ)2L2(Ω), (4.21)
    ϱ(ΔU,Uttt)L2(Qτ)=ϱ(U(x,τ),Uττ(x,τ))L2(Ω)ϱ2Uτ(x,τ)2L2(Ω)ϱΩUττ(x,τ)(τ0ΩU(ξ,η)dξdη)dsx+ϱΩτ0Utt(x,t)ΩU(ξ,t)dξdtdsx, (4.22)
    δ(ΔUt,Uttt)L2(Qτ)=δ(Uτ(x,τ),Uττ(x,τ))L2(Ω)δτ0Utt(x,)2L2(Ω)dtδΩUττ(x,τ)ΩU(ξ,τ)dξdsx+δτ0ΩUtt(x,t)ΩUt(ξ,t)dξdsxdt, (4.23)
    γ(ΔUtt,Uttt)L2(Qτ)=γ2Uττ(x,τ)2L2(Ω)γΩUττ(x,τ)ΩUτ(ξ,τ)dξdsx+γτ0ΩUtt(x,t)ΩUtt(ξ,t)dξdsxdt. (4.24)
    (ΔW,Uttt)L2(Qτ)=H(τ)(U(x,τ),Uττ(x,τ))L2(Ω)+ΩUτττ0h(τσ)(U(τ)U(σ))dσdxτ0(Utt,t0h(tσ)(U(t)U(σ))dσ)L2(Ω)dt+τ0h(t)(Utt,U(t))L2(Ω)dt+ΩUττ(x,τ)(τ0ΩW(ξ,η)dξdη)dsxΩτ0Utt(x,t)ΩW(ξ,t)dξdtdsx, (4.25)

    Substitution (4.19)-(4.25) into (4.18) we get the equality

    12Uτττ(x,τ)2L2(Ω)+β2Uττ(x,τ)2L2(Ω)+ϱ(U(x,τ),Uττ(x,τ))L2(Ω)+δ(Uτ(x,τ),Uττ(x,τ))L2(Ω)+γ2Uττ(x,τ)2L2(Ω)ϱ2Uτ(x,τ)2L2(Ω)H(τ)(U(x,τ),Uττ(x,τ))L2(Ω)+ΩUτττ0h(τσ)(U(τ)U(σ))dσdx=ατ0Uttt(x,t)2L2(Ω)dt+δτ0Utt(x,)2L2(Ω)dt+ϱΩUττ(x,τ)(τ0ΩU(ξ,η)dξdη)dsxϱΩτ0Utt(x,t)ΩU(ξ,t)dξdtdsx+δΩUττ(x,τ)ΩU(ξ,τ)dξdsxδτ0ΩUtt(x,t)ΩUt(ξ,t)dξdsxdt+γΩUττ(x,τ)ΩUτ(ξ,τ)dξdsxγτ0ΩUtt(x,t)ΩUtt(ξ,t)dξdsxdtτ0(Utt,t0h(tσ)(U(t)U(σ))dσ)L2(Ω)dt+τ0h(t)(Utt,U(t))L2(Ω)dt+ΩUττ(x,τ)(τ0ΩW(ξ,η)dξdη)dsxΩτ0Utt(x,t)ΩW(ξ,t)dξdtdsx. (4.26)

    The right hand side of (4.26) can be bounded as follows

    ϱΩUττ(x,τ)(τ0ΩU(ξ,η)dξdη)dsxϱ2ε1(εUττ(x,τ)2L2(Ω)+l(ε)Uττ(x,τ)2L2(Ω))+ϱ2ε1T|Ω||Ω|τ0U(x,t)2L2(Ω)dt, (4.27)
    ϱΩτ0Utt(x,t)ΩU(ξ,t)dξdtdsxϱ2τ0{εUtt(x,t)2L2(Ω)+l(ε)Utt(x,t)2L2(Ω)}dt+ϱ2|Ω||Ω|τ0U(x,t)2L2(Ω)dt, (4.28)
    δΩUττ(x,τ)ΩU(ξ,τ)dξdsxδ2ε2(εUττ(x,τ)2L2(Ω)+l(ε)Uττ(x,τ)2L2(Ω))+δ2ε2T|Ω||Ω|U(x,τ)2L2(Ω), (4.29)
    δτ0ΩUtt(x,t)ΩUt(ξ,t)dξdsxdtδ2ετ0Utt(x,t)2L2(Ω)dt+δ2l(ε)τ0Utt(x,t)2L2(Ω)dt+δ2T|Ω||Ω|τ0Ut(x,t)2L2(Ω)dt, (4.30)
    γΩUττ(x,τ)ΩUτ(ξ,τ)dξdsxγ2ε3(εUττ(x,τ)2L2(Ω)+l(ε)Uττ(x,τ)2L2(Ω))+γ2ε3T|Ω||Ω|Uτ(x,τ)2L2(Ω), (4.31)
    γτ0ΩUtt(x,t)ΩUtt(ξ,t)dξdsxdtγ2l(ε)τ0Utt(x,t)2L2(Ω)dt+γ2ετ0Utt(x,t)2L2(Ω)dt+γ2T|Ω||Ω|τ0Utt(x,t)2L2(Ω)dt, (4.32)
    ΩUττ(x,τ)(τ0ΩW(ξ,η)dξdη)dsx(h02ε6+12ε7)(εUττ(x,τ)2L2(Ω)+l(ε)Uττ(x,τ)2L2(Ω))+h02ε6T|Ω||Ω|τ0U(x,t)2L2(Ω)dt+12ε7|Ω||Ω|τ0hU(t)dt, (4.33)
    Ωτ0Utt(x,t)ΩW(ξ,t)dξdtdsxh0+12τ0{εUtt(x,t)2L2(Ω)+l(ε)Utt(x,t)2L2(Ω)}dt+h02|Ω||Ω|τ0U(x,t)2L2(Ω)dt+12|Ω||Ω|τ0hU(t)dt. (4.34)
    ΩUτττ0h(τσ)(U(τ)U(σ))dσdx12ε8h0Uττ(x,τ)2L2(Ω)12ε8hU(τ), (4.35)
    H(τ)(U(x,τ),Uττ(x,τ))L2(Ω)12ε9h0Uττ(x,τ)2L2(Ω)12ε9h0U(x,τ)2L2(Ω), (4.36)
    τ0h(t)Utt(x,t)U(x,t)dth(0)2τ0Utt(x,τ)2L2(Ω)dt+h(0)2τ0U(x,τ)2L2(Ω)dt, (4.37)
    τ0Uttt0h(tσ)(U(t)U(σ))dσdxh(t)h(0)2τ0Utt(x,τ)2L2(Ω)dt12τ0hU(t)dt. (4.38)

    So, combining inequalities (4.27)-(4.38) and equality (4.26) we obtain

    12Uτττ(x,τ)2L2(Ω)+{β2ϱ2ε1l(ε)δ2ε2l(ε)γ2ε3l(ε)(h02ε6+12ε7)l(ε)}Uττ(x,τ)2L2(Ω)γ2ε3T|Ω||Ω|Uτ(x,τ)2L2(Ω)δ2ε2T|Ω||Ω|U(x,τ)2L2(Ω)+{γ2ϱ2ε1εδ2ε2εγ2ε3ε+(h02ε6+12ε7)ε(12ε8+12ε9)h0}Uττ(x,τ)2L2(Ω)ε82hU(τ)ε92h0U(x,τ)2L2(Ω)ϱ2Uτ(x,τ)2L2(Ω)+ϱ(U(x,τ),Uττ(x,τ))L2(Ω)+δ(Uτ(x,τ),Uττ(x,τ))L2(Ω)ατ0Uttt(x,t)2L2(Ω)dt+{ϱ2l(ε)+δ2l(ε)+γ2l(ε)+γ2T|Ω||Ω|+(h0+10)l(ε)}τ0Utt(x,t)2L2(Ω)dt+{ϱ2ε1T|Ω||Ω|+(ϱ2+h02(1+Tε6))|Ω||Ω|}τ0U(x,t)2L2(Ω)dt+δ2T|Ω||Ω|τ0Ut(x,t)2L2(Ω)dt+{δ+ϱ2ε+δ2ε+γ2ε+h0+12ε+3h(0)2}τ0Utt(x,)2L2(Ω)dt12τ0hU(t)dth(0)2τ0U(x,τ)2L2(Ω)dt+1+ε72|Ω||Ω|τ0hU(t)dt. (4.39)

    Adding side to side (4.17) and (4.39), we obtain

    \begin{equation} \begin{array}{l} \left\{ \frac{\beta }{2}-\frac{\delta }{2}\varepsilon _{2}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert-\frac{1+\alpha}{2} \right\} \left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{1}{ 2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\\ +\left\{- \frac{1+\alpha}{2}-\frac{\gamma }{2}\varepsilon _{3}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\beta }{2}-\frac{\varrho }{2\varepsilon _{1}^{\prime }} l(\varepsilon )-l(\varepsilon )\frac{\delta }{2\varepsilon _{2}^{\prime }}- \frac{\gamma }{2\varepsilon _{3}^{\prime }}l(\varepsilon )-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})-\frac{1}{2}\right\} \left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +(U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }+\alpha \left( U_{\tau }(x, \tau ), U(x, \tau )\right) _{L^{2}(\Omega )}+\frac{\varrho }{2}\left\Vert \nabla v(x, 0)\right\Vert _{L^{2}(\Omega )}^{2} \\ +\varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}+\delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )} \\ +(\frac{\gamma }{2}-\frac{\varepsilon'_{9}}{2}h_{0})\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varrho }{2}\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varepsilon'_{8}}{2}h\circ \nabla U(\tau)\\ +\left\{ \frac{\gamma }{2}-\frac{\varrho }{ 2\varepsilon _{1}^{\prime }}\varepsilon -\frac{\delta }{2\varepsilon _{2}^{\prime }}-\frac{\gamma }{2\varepsilon _{3}^{\prime }}\varepsilon-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})\varepsilon-(\frac{1}{2\varepsilon _{8}^{\prime }}+\frac{1}{2\varepsilon _{9}^{\prime }})h_{0} \right\} \left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ \leq \left\{ \frac{\varrho }{2}\varepsilon _{1}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert +\frac{\varrho }{2} \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{ \varrho }{2}T^{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\delta }{2}\left( T^{2}l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \right. \\ \left. +\frac{\gamma }{2}l(\varepsilon )T^{2}+\frac{h_{0}}{2}T^{2}(l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert)+\frac{1}{2}l(\varepsilon)\right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left( \alpha +\frac{\gamma \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert }{2}+\frac{\delta }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \frac{1}{2}+l(\varepsilon )\frac{\varrho }{2}+\frac{\delta }{2} l(\varepsilon )+\frac{\gamma }{2}l(\varepsilon )+\frac{\gamma }{2} T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right\} \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \frac{\delta }{2}\varepsilon +\frac{\gamma }{2}\varepsilon +\varepsilon \frac{\varrho }{2}+\delta+\frac{h_{0}+1}{2}\varepsilon+\frac{3h(0)}{2} \right\} \int_{0}^{\tau }\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ -\alpha \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\bigg(h_{0}+\left( \frac{\varrho +\delta +\gamma+h_{0} }{2}\right) \varepsilon\bigg) \int_{0}^{\tau }\left\Vert \nabla v\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\frac{h_{0}+h(0)}{2}\int_{0}^{\tau}\left\Vert \nabla U(x, t )\right\Vert _{L^{2}(\Omega )}^{2}dt+\frac{1}{2}\int_{0}^{\tau}h\circ \nabla U(t )dt+\frac{1}{2}\vert \Omega\vert\vert \partial \Omega \vert\int_{0}^{\tau}h\circ U(t )dt\\ -\frac{1}{2}\int_{0}^{\tau }h'\circ \nabla U(t)dt+\frac{1+\varepsilon'_{7}}{2}\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert\int_{0}^{\tau }h'\circ U(t)dt. \end{array} \end{equation} (4.40)

    Now to deal with the last term on the right hand side of (4.40) , we define the function \theta \left(x, t\right) by the relation

    \begin{equation*} \theta \left( x, t\right) : = \int_{0}^{t}U(x, s)ds. \end{equation*}

    Hence using (4.12) it follows that

    \begin{equation} v\left( x, t\right) = \theta (x, \tau )-\theta \left( x, t\right) , \ \nabla v(x, 0) = \nabla \theta (x, \tau ), \end{equation} (4.41)

    and

    \begin{eqnarray} \left\Vert \nabla v\right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2} & = &\left\Vert \nabla \theta (x, \tau )-\nabla \theta \left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &\leq &2\left( \tau \left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla \theta \left( x, t\right) \right\Vert _{L^{2}\left( Q_{\tau }\right) }^{2}\right) . \end{eqnarray} (4.42)

    And make use of the following inequality

    \begin{equation} -\frac{\alpha }{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\alpha }{2}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq \alpha (U_{\tau }(x, \tau ), U(x, \tau ))_{L^{2}(\Omega )}, \end{equation} (4.43)
    \begin{equation} -\frac{1}{2}\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{1}{2}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq (U_{\tau \tau }(x, \tau ), U(x, \tau ))_{L^{2}\left( \Omega \right) }, \end{equation} (4.44)
    \begin{equation} -\frac{\varrho }{2\varepsilon _{4}^{\prime }}\left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\varrho }{2}\varepsilon _{4}^{\prime }\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq \varrho \left( \nabla U(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}, \end{equation} (4.45)
    \begin{equation} -\frac{\delta }{2\varepsilon _{5}^{\prime }}\left\Vert \nabla U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}-\frac{\delta }{2}\varepsilon _{5}^{\prime }\left\Vert \nabla U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}\leq \delta \left( \nabla U_{\tau }(x, \tau ), \nabla U_{\tau \tau }(x, \tau )\right) _{L^{2}(\Omega )}. \end{equation} (4.46)
    \begin{eqnarray} m_{1}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{1}\left\Vert U(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{1}\left\Vert U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2} , \\ m_{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{2}\left\Vert U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{2}\left\Vert U_{tt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}, \\ m_{3}\left\Vert U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{3}\left\Vert U_{tt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{3}\left\Vert U_{ttt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2} , \\ m_{4}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{4}\left\Vert \nabla U(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{4}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}, \end{eqnarray} (4.47)
    \begin{eqnarray} m_{5}\left\Vert \nabla U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}&\leq& m_{5}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+m_{5}\left\Vert \nabla U_{tt}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}, \\ m_{6}h\circ \nabla U (\tau)&\leq&m_{6}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+ m_{6}\int_{0}^{\tau}h\circ\nabla U (t)dt\\ m_{7}h\circ U (\tau)&\leq&m_{7}\left\Vert U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}+ m_{7}\int_{0}^{\tau}h\circ U (t)dt\\ -m_{8}h'\circ \nabla U (\tau)&\leq&m_{8}\left\Vert \nabla U_{t}(x, t)\right\Vert _{L^{2}(Q_{\tau })}^{2}- m_{8}\int_{0}^{\tau}h'\circ\nabla U (t)dt. \end{eqnarray} (4.48)

    Let

    \begin{equation} \left\{ \begin{array}{l} m_{1}: = \frac{1+\alpha}{2}+\frac{\delta }{2}\varepsilon _{2}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert, \\ \\ m_{2}: = 1+\frac{\gamma }{2}\varepsilon _{3}^{\prime }T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\alpha }{2} \\ \\ m_{3}: = \bigg(\frac{\varrho }{2\varepsilon _{1}^{\prime }}+\frac{\delta }{2\varepsilon _{2}^{\prime }}+\frac{\gamma }{ 2\varepsilon _{3}^{\prime }}+\frac{h_{0} }{2\varepsilon _{6}^{\prime }}+\frac{1 }{2\varepsilon _{7}^{\prime }}\bigg)l(\varepsilon )+\frac{1}{2} \\ \\ m_{4}: = \frac{\varrho }{2}\varepsilon _{4}^{\prime }+\frac{h_{0}}{2} \\ m_{5}: = 1+\frac{\varrho }{2}+\frac{\delta }{2\varepsilon _{5}^{\prime }}\\ m_{6}: = \frac{1}{2}\varepsilon _{8}^{\prime }+1 , \quad m_{7}: = 1, \quad m_{8}: = 1, \end{array} \right. \end{equation} (4.49)

    choosing \varepsilon _{1}^{\prime }, \ \varepsilon _{2}^{\prime }, \varepsilon _{3}^{\prime }, \varepsilon _{4}^{\prime } , \varepsilon _{5}^{\prime }, \ \varepsilon _{6}^{\prime }, \varepsilon _{7}^{\prime }, \varepsilon _{8}^{\prime } and \varepsilon _{9}^{\prime } sufficiently large

    \begin{equation} \alpha_{0}: = \frac{\gamma }{2}-\frac{\varrho }{ 2\varepsilon _{1}^{\prime }}\varepsilon -\frac{\delta }{2\varepsilon _{2}^{\prime }}-\frac{\gamma }{2\varepsilon _{3}^{\prime }}\varepsilon-\frac{\varrho }{2\varepsilon _{4}^{\prime }}-\frac{\delta }{2\varepsilon _{5}^{\prime }}\varepsilon-(\frac{h_{0}}{2\varepsilon _{6}^{\prime }}+\frac{1}{2\varepsilon _{7}^{\prime }})\varepsilon-(\frac{1}{2\varepsilon _{8}^{\prime }}+\frac{1}{2\varepsilon _{9}^{\prime }})h_{0} > 0. \end{equation} (4.50)

    Since \tau is arbitrary we get that \alpha_{1}: = \frac{\varrho }{2}-2\tau \bigg(h_{0}+\varepsilon \frac{\left(\varrho +\delta +\gamma+h_{0} \right)}{2}\bigg) > 0, \ thus inequality (4.40) takes the form

    \begin{equation} \begin{array}{l} \frac{\beta }{2}\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{ 1}{2}\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{ \beta }{2}\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{1}{2}\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\{ \frac{\varrho }{2}-2\tau(h_{0}+ \varepsilon\frac{\left( \varrho +\delta +\gamma+h_{0} \right)}{2}) \right\} \left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\frac{\gamma }{2}\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\alpha_{0} \left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} +h\circ \nabla U(\tau)+h\circ U(\tau)-h'\circ \nabla U(\tau)\\ \leq \left\{ \gamma _{1}^{\prime }+m_{1}\right\} \int_{0}^{\tau }\left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\left( \gamma _{2}^{\prime }+m_{1}+m_{2}+m_{7}\right) \int_{0}^{\tau }\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +\left\{ \gamma _{3}^{\prime }+m_{2}+m_{3}\right\} \int_{0}^{\tau }\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+\left( m_{3}-\alpha \right) \int_{0}^{\tau }\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +(2h_{0}+ \varepsilon\left( \varrho +\delta +\gamma+h_{0} \right)) \int_{0}^{\tau }\left\Vert \nabla \theta \left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+(\gamma _{4}^{\prime } +m_{5}) \int_{0}^{\tau }\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt \\ +(m_{4}+\frac{h_{0}+h(0)}{2})\int_{0}^{\tau }\left\Vert \nabla U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt\\ +\left( m_{4}+m_{5}+m_{6}+m_{8}\right) \int_{0}^{\tau }\left\Vert \nabla U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}dt+(\frac{1}{2}+m_{6})\int_{0}^{\tau }h\circ \nabla U(t)dt \\ +(\gamma'_{5}+m_{7})\int_{0}^{\tau }h\circ U(t)dt-(\frac{1}{2}+m_{8})\int_{0}^{\tau }h'\circ \nabla U(t)dt, \end{array} \end{equation} (4.51)

    where

    \begin{equation} \left\{ \begin{array}{l} \gamma _{1}^{\prime }: = \frac{\varrho }{2}\varepsilon _{1}^{\prime }T\left\vert \partial \Omega \right\vert \left\vert \Omega \right\vert + \frac{\varrho }{2}\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert +\frac{\varrho }{2}T^{2}\left( l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right)\\ \quad +\frac{\delta }{2} \left( T^{2}l(\varepsilon )+\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \right) +\frac{\gamma }{2}l(\varepsilon )T^{2} \\ \\ \gamma _{2}^{\prime }: = \alpha +\frac{\gamma \left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert }{2}+\frac{\delta }{2}T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \\ \gamma _{3}^{\prime }: = \frac{1}{2}+l(\varepsilon )\frac{\varrho }{2}+\frac{ \delta }{2}l(\varepsilon )+\frac{\gamma }{2}l(\varepsilon )+\frac{\gamma }{2} T\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert\\ \gamma _{4}^{\prime }: = \frac{\delta }{2}\varepsilon +\frac{\gamma }{2}\varepsilon +\varepsilon \frac{\varrho }{2}+\delta \\ \gamma _{5}^{\prime }: = (1+\frac{1}{2}\varepsilon'_{7})\left\vert \Omega \right\vert \left\vert \partial \Omega \right\vert \end{array} \right. \end{equation} (4.52)

    We obtain

    \begin{eqnarray} &&\left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2} \\ &&+\left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+h\circ \nabla U(\tau)+h\circ U(\tau)-h'\circ \nabla U(\tau)\\ &\leq &D\int_{0}^{\tau }\bigg\{ \left\Vert U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+h\circ U(t) \\ &&+\left\Vert \nabla U\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{t}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{tt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}-h'\circ \nabla U(t) \\ && +\left\Vert U_{ttt}\left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla \theta \left( x, t\right) \right\Vert _{L^{2}(\Omega )}^{2}+h\circ \nabla U(t)\bigg\} dt, \end{eqnarray} (4.53)

    where

    \begin{equation} D: = \frac{ \begin{array}{c} \max \left\{ (\gamma _{1}^{\prime }+m_{1}), \left( \gamma _{2}^{\prime }+m_{1}+m_{2}+m_{7}\right), \gamma _{3}^{\prime }+m_{2}+m_{3}, m_{3}-\alpha , \right. \\ \left. m_{4}+m_{5}+m_{6}+m_{8}, \gamma _{4}^{\prime } +m_{5}, (2h_{0}+\varepsilon \left( \varrho +\delta +\gamma+h_{0} \right)), \right. \\ \left.m_{4}+\frac{h_{0}+h(0)}{2}, \frac{1}{2}+m_{6}, \gamma'_{5}+m_{7}, \frac{1}{2}+m_{8} \right\} \end{array} }{ \begin{array}{c} \min \left\{ \frac{\beta }{2}, \frac{1}{2}, , \frac{\gamma }{2}, \alpha_{0} , \alpha_{1} \right\} \end{array} }. \end{equation} (4.54)

    Further, applying Gronwall's lemma to (4.53), we deduce that

    \begin{equation} \begin{array}{l} \left\Vert U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert U_{\tau \tau \tau }(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2} \\ +\left\Vert \nabla U(x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}+\left\Vert \nabla U_{\tau \tau }\left( x, \tau \right) \right\Vert _{L^{2}(\Omega )}^{2}\\ +\left\Vert \nabla \theta (x, \tau )\right\Vert _{L^{2}(\Omega )}^{2}+h\circ \nabla U(\tau)+h\circ U(\tau)-h'\circ \nabla U(\tau) \leq 0, \forall \tau \in \left[ 0, \alpha_{2}\right] . \end{array} \end{equation} (4.55)

    where \alpha_{2}: = \frac{\varrho }{4h_{0}+2\varepsilon \left(\varrho +\delta +\gamma+h_{0} \right) } .

    Proceeding in the same way for the intervals \tau \in \left[(m-1)\alpha_{2}, m\alpha_{2}\right] \ to cover the whole interval \left[0, T\right], and thus proving that U(x, \tau) = 0 , for all \tau in \left[0, T\right].\ Thus, the uniqueness is proved.

    Study of sound wave propagation, it should be noted that the Moore-Gibson-Thomson equation is one of the nonlinear sound equations that describes the propagation of sound waves in gases and liquids. The behavior of sound waves depends strongly on the average scattering, scattering and nonlinear effects. Arises from high-frequency ultrasound (HFU) modeling (see [16,25,41]). In this work, we have studied the solvability of the nonlocal mixed boundary value problem for the fourth order of Moore-Gibson-Thompson equation with source and memory terms. Galerkin's method was the main used tool for proving the solvability of the given non local problem. In the next work, we will try to using the same method with Hall-MHD equations which are nonlinear partial differential equation that arises in hydrodynamics and some physical applications (see for example [2,3,4,6]) by using some famous algorithms (see [8,14,15]).

    The fourth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P-2/1/42).

    This work does not have any conflicts of interest.

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