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Research article

Mathematical modeling of HIV/HTLV co-infection with CTL-mediated immunity

  • In the literature, a great number of HIV and HTLV-I mono-infection models has been formulated and analyzed. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In the present paper we formulate and analyze a new HIV/HTLV-I co-infection model with latency and Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4+T cells, latently HIV-infected cells, actively HIV-infected cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell and cell-to-cell transmissions, while the HTLV-I can only spread via cell-to-cell transmission. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we have discussed the effect of HTLV-I infection on the HIV-infected patients and vice versa. We have pointed out the influence of CTL immune response on the co-infection dynamics.

    Citation: A. M. Elaiw, N. H. AlShamrani, A. D. Hobiny. Mathematical modeling of HIV/HTLV co-infection with CTL-mediated immunity[J]. AIMS Mathematics, 2021, 6(2): 1634-1676. doi: 10.3934/math.2021098

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  • In the literature, a great number of HIV and HTLV-I mono-infection models has been formulated and analyzed. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In the present paper we formulate and analyze a new HIV/HTLV-I co-infection model with latency and Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4+T cells, latently HIV-infected cells, actively HIV-infected cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell and cell-to-cell transmissions, while the HTLV-I can only spread via cell-to-cell transmission. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we have discussed the effect of HTLV-I infection on the HIV-infected patients and vice versa. We have pointed out the influence of CTL immune response on the co-infection dynamics.


    The fundamental work of Hansen and Spies [4] modeled a two-layer beam with a structural damping due to the interfacial slip through the following system

    {ρφtt+G(ψφx)x=0,Iρ(3wψ)ttD(3wψ)xxG(ψφx)=0,IρwttDwxx+3G(ψφx)+4γw+4βwt=0, (1.1)

    where φ=φ(x,t) is the transverse displacement, ψ=ψ(x,t) is the rotation angle, w=w(x,t) is proportional to the amount of slip along the interface, 3wψ denotes the effective rotation angle. The physical quantities ρ,Iρ,G,D,β and γ are respectively: the density, mass moment of inertia, shear stiffness, flexural rigidity, adhesive damping and adhesive stiffness. Equation (1.1)3 describes the dynamics of the slip. For β=0, system (1.1) describes the coupled laminated beams without structural damping at the interface. In the recent result [1], Apalara considered the thermoelastic-laminated beam system without structural damping, namely

    {ρφtt+G(ψφx)x=0,Iρ(3sψ)ttD(3sψ)xxG(ψφx)=0,IρsttDsxx+3G(ψφx)+4γs+δθx=0,ρ3θtλθxx+δstx=0, (1.2)

    where (x,t)(0,1)×(0,+), θ=θ(x,t) is the difference temperature. The positive quantities γ,β,k,λ are adhesive stiffness, adhesive damping, heat capacity and the diffusivity respectively. The author proved that (1.2) is exponential stable provided

    Gρ=DIρ. (1.3)

    When β>0, the adhesion at the interface supplies a restoring force proportion to the interfacial slip. But this is not enough to stabilize system (1.1), see for instance [2]. To achieve exponential or general stabilization of system (1.1), many authors in literature have used additional damping. In this direction, Gang et al. [9] studied the following memory-type laminated beam system

    {ρφtt+G(ψφx)x=0,Iρ(3wψ)ttD(3wψ)xx+t0g(ts)(3wψ)xx(x,s)dsG(ψφx)=03Iρwtt3Dwxx+3G(ψφx)+4γw+4βwt=0 (1.4)

    and established a general decay result for more regular solutions and GρDIρ. Mustafa [15] also considered the structural damped laminated beam system (1.4) and established a general decay result provided Gρ=DIρ. Feng et al. [8] investigated the following laminated beam system

    {ρwtt+Gφx+g1(wt)+f1(w,ξ,s)=h1,IρξttGφDξxx+g2(ξt)+f2(w,ξ,s)=h2,Iρstt+GφDsxx+g3(st)+f2(w,ξ,s)=h3 (1.5)

    and established the well-posedness, smooth global attractor of finite fractal dimension as well as existence of generalized exponential attractors. See also, recent results by Enyi et al. [20]. We refer the reader to [5,6,7,11,13,14,17,18] and the references cited therein for more related results.

    In this present paper, we consider a thermoelastic laminated beam problem with a viscoelastic damping

    {ρwtt+G(ψwx)x=0,Iρ(3sψ)ttD(3sψ)xx+t0g(tτ)(3sψ)xx(x,τ)dτG(ψwx)=03Iρstt3Dsxx+3G(ψwx)+4γs+δθx=0,kθtλθxx+δsxt=0 (1.6)

    under initial conditions

    {w(x,0)=w0(x), ψ(x,0)=ψ0(x), s(x,0)=s0(x), θ(x,0)=θ0(x),  x[0,1],wt(x,0)=w1(x), ψt(x,0)=ψ1(x), st(x,0)=s1(x),                          x[0,1] (1.7)

    and boundary conditions

    {w(0,t)=ψx(0,t)=sx(0,t)=θ(0,t)=0,t[0,+),wx(1,t)=ψ(1,t)=s(1,t)=θx(1,t)=0,t[0,+). (1.8)

    In the system (1.6), the integral represents the viscoelastic damping, and g is the relaxation function satisfying some suitable assumptions specified in the next section. According to the Boltzmann Principle, the viscoelastic damping (see [21] for details) is represented by a memory term in the form of convolution. It acts as a damper to reduce the internal/external forces like the beam's weight, heavy loads, wind, etc., that cause undesirable vibrations.

    In most of the above works, the authors have established their decay result by including the structural damping along with other dampings. So, the natural question that comes to mind.

    Is it possible to obtain general/optimal decay result (decay rates that agrees with that of g) to the thermoelastic laminated beam system (1.6)–(1.8), in the absence of the structural damping.

    The novelty of this article is to answer this question in a consenting way, by using the ideas developed in [10] to establish general and optimal decay results for Problem 1.6. Moreover, we establish a weaker decay result in the case of a non-equal wave of speed propagation. To the best of our knowledge, there is no stability result for the latter in the literature.

    The rest of work is organized as follows: In Section 2, we recall some preliminaries and assumptions on the memory term. In Section 3, we state and prove the main stability result for the case equal-speed and in the case of non-equal-speed of propagation. We also give some examples to illustrate our findings. Finally, in Section 4, we give the proofs of the lemmas used our main results.

    In this section, we recall some useful materials and conditions. Through out this paper, C is a positive constant that may change through lines, .,. and .2 denote respectively the inner product and the norm in L2(0,1). We assume the relaxation function g obeys the assumptions:

    (G1). g:[0,+)(0,+) is a non-inecreasing C1 function such that

    g(0)>0,D0g(τ)dτ=l0>0. (2.1)

    (G2). There exist a C1 function H:[0,+)(0,+) which is linear or is strictly convex C2 function on (0,ϵ0), ϵ0g(0), with H(0)=H(0)=0 and a positive nonincreasing differentiable function ξ:[0,+)(0,+), such that

    g(t)ξ(t)H(g(t)),t0, (2.2)

    Remark 2.1. As in [10], we note here that, if H is a strictly increasing convex C2 function on (0,r], with H(0)=H(0)=0, then H has an extension ˉH, which is strictly increasing and strictly convex C2-function on (0,+). For example, ˉH can be defined by

    ˉH(s)=H(r)2s2+(H(r)H(r)r)s+H(r)H(r)r+H(r)2r2, s>r. (2.3)

    Let

    H1(0,1)={uH1(0,1)/u(0)=0},   ˉH1(0,1)={uH1(0,1)/u(1)=0},
    H2(0,1)={uH2(0,1)/uxH1(0,1)},   ˉH2(0,1)={uH2(0,1)/uxˉH1(0,1)}.

    The existence and regularity result of problem (1.6) is the following

    Theorem 2.1. Let (w0,3s0ψ0,s0,θ0)H1(0,1)×ˉH1(0,1)×ˉH1(0,1)×H1(0,1) and (w1,3s1ψ1,s1)L2(0,1)×L2(0,1)×L2(0,1) be given. Suppose (G1) and (G2) hold. Then problem (1.6) has a unique global weak solution (w,3sψ,s,θ) which satisfies

    wC(R+,H1(0,1))C1(R+,L2(0,1)), (3sψ)C(R+,ˉH1(0,1))C1(R+,L2(0,1)),
    sC(R+,ˉH1(0,1))C1(R+,L2(0,1)), θC(R+,L2(0,1))L2(R+,H1(0,1)).

    Furthermore, if (w0,(3s0ψ0),s0,θ0)H2(0,1)×ˉH2(0,1)×ˉH2(0,1)×H2(0,1)H1(0,1) and (w1,(3s1ψ1),s1)H1(0,1)×ˉH1(0,1)×ˉH1(0,1), then the solution of (1.6) satisfies

    wC(R+,H2(0,1))C1(R+,H1(0,1))C2(R+,L2(0,1)),
    (3sψ)C(R+,ˉH2(0,1))C1(R+,ˉH1(0,1))C2(R+,L2(0,1)),
    sC(R+,ˉH2(0,1))C1(R+,ˉH1(0,1))C2(R+,L2(0,1)),
    θC(R+,H2(0,1)H2(0,1))C1(R+,H1(0,1)).

    The proof of Theorem 2.1 can be established using the Galerkin approximation method as in [16]. Throughout this paper, we denote by the binary operator, defined by

    (gν)(t)=t0g(tτ)ν(t)ν(τ)22dτ,t0.

    We also define h(t) and Cα as follow

    h(t)=αg(t)g(t)   and   Cα=+0g2(τ)αg(τ)g(τ)dτ.

    The following lemmas will be applied repeatedly throughout this paper

    Lemma 2.1. For any function fL2loc([0,+),L2(0,1)), we have

    10(t0g(ts)(f(t)f(s))ds)2dx(1l0)(gf)(t), (2.4)
    10(x0f(y,t)dy)2dxf(t)22. (2.5)

    Lemma 2.2. Let vH1(0,1)  or  ˉH1(0,1), we have

    10(t0g(ts)(v(t)v(τ))dτ)2dxCp(1l0)(gv)(t), (2.6)

    where Cp>0 is the poincaré constant.

    Lemma 2.3. Let (w,3sψ,s,θ) be the solution of (1.6). Then, for any 0<α<1 we have

    10(t0g(tτ)((3sψ)x(τ)(3sψ)x(t))dτ)2dxCα(h(3sψ)x)(t). (2.7)

    Proof. Using Cauchy-Schwarz inequality, we have

    10(t0g(tτ)((3sψ)x(τ)(3sψ)x(t))dτ)2dx=10(t0g(tτ)h(tτ)h(tτ)((3sψ)x(τ)(3sψ)x(t))dτ)2dx(+0g2(τ)h(τ)ds)10t0h(tτ)((3sψ)x(τ)(3sψ)x(t))2dτdx=Cα(h(3sψ)x)(t). (2.8)

    Lemma 2.4. [12] Let F be a convex function on the close interval [a,b], f,j:Ω[a,b] be integrable functions on Ω, such that j(x)0 and Ωj(x)dx=α1>0. Then, we have the following Jensen inequality

    F(1α1Ωf(y)j(y)dy)1α1ΩF(f(y))j(y)dy. (2.9)

    In particular if F(y)=y1p,  y0,  p>1, then

    (1α1Ωf(y)j(y)dy)1p1α1Ω(f(y))1pj(y)dy. (2.10)

    Lemma 2.5. The energy functional E(t) of the system (1.6)-(1.8) defined by

    E(t)=12[ρwt22+3Iρst22+Iρ3stψt22+3Dsx22+Gψwx22]+12[(Dt0g(τ)dτ)3sxψx22+(g(3sxψx))(t)+4γs22+kθ22], (2.11)

    satisfies

    E(t)=12(g(3sxψx))(t)12g(t)3sxψx22λθx2212(g(3sxψx))(t)0,   t0. (2.12)

    Proof. Multiplying (1.6)1, (1.6)2, (1.6)3 and (1.6)4, respectively, by wt, (3stψt), st and θ, integrating over (0,1), and using integration by parts and the boundary conditions (1.7), we arrive at

    12ddt(ρwt22+Gψwx22)=G(ψwx),ψt, (2.13)
    12ddt[Iρ3stψt22+(Dt0g(τ)dτ)3sxψx22+(g(3sxψx))(t)]=G(ψwx),(3sψ)t+12(g(3sxψx))(t)12g(t)3sxψx22, (2.14)
    12ddt[3Iρst22+3Dsx22+4γs22]=3G(ψwx),stδθx,st, (2.15)

    and

    12ddt(kθ22)=λθx22+δθx,st. (2.16)

    Adding the equations (2.13)–(2.16), taking into account (G1) and (G2), we obtain (2.12) for regular solutions. The result remains valid for weak solutions by a density argument. This implies the energy functional is non-increasing and

    E(t)E(0),  t0.

    This section is subdivided into two. In the first subsection, we prove the stability result for equal-wave-speed of propagation, whereas in the second subsection, we focus on the stability result for non-equal-wave-speed of propagation.

    Our aim, in this subsection, is to prove an explicit, general and optimal decay rate of solutions for system (1.6)–(1.8). To achieve this, we define a Lyapunov functional

    L(t)=NE(t)+6j=1NjIj(t), (3.1)

    where N, Nj, j=1,2,3,4,5,6 are positive constants to be specified later and

    I1(t)=Iρ10(3sψ)tt0g(tτ)((3sψ)(t)(3sψ)(τ))dτdx,t0,
    I2(t)=3Iρ10sstdx+3ρ10wtx0s(y)dydx,I3(t)=3kIρ10θx0st(y)dydx,t0,
    I4(t)=ρ10wtwdx,I5(t)=Iρ10(3sψ)(3sψ)tdx,t0,
    I6(t)=3IρG10(ψwx)stdx3ρD10wtsxdx,I7(t)=10t0J(tτ)(3sxψx)2(τ)dτdx,t0,

    where

    J(t)=+tg(τ)dτ.

    The following lemma is very important in the proof of our stability result.

    Lemma 3.1. Suppose Gρ=DIρ. Under suitable choice of t0,N, Nj, j=1,2,3,4,5,6, the Lyapunov functional L satisfies, along the solution of (1.6)(1.8), the estimate

    L(t)β(wt22+st22+3stψt22+sx22+3wxψx22+ψwx22)β(s22+θx22)+12(g(3sxψx))(t), tt0 (3.2)

    and the equivalence relation

    α1E(t)L(t)α2E(t) (3.3)

    holds for some β>0, α1, α2>0.

    Proof. By virtue of assumption (3.1) and using h(t)=αg(t)g(t), it follows from Lemmas 2.5, 4.1-4.6 (see the Appendix for detailed derivations) that, for all tt0>0,

    L(t)[N4ρN2δ4]wt22[N3δIρ2N2C(1+1ϵ2)N6C(1+1ϵ1)]st223N2γs22[N1Iρg0N5IρN6ϵ1]3stψt22[3DN2N3ϵ3N4CN6C]sx22[N6G2N1ϵ2N3ϵ3N4Cϵ4N5C]ψwx22[N5l04N1ϵ1N4ϵ4]3sxψx22[λNN2CN3C(1+1ϵ3)N6C]θx22+Nα2(g(3sxψx))(t)[N2CCα(N5+N1(1+1ϵ1+1ϵ2))](h(3sxψx))(t). (3.4)

    Now, we choose

    N4=N5=1,  ϵ4=l08 (3.5)

    and select N1 large enough such that

    μ1:=N1Iρg0Iρ>0. (3.6)

    Next, we choose N6 large so that

    μ2:=N6G2C>0. (3.7)

    Also, we select N2 large enough so that

    μ3:=3DN2CN6C>0. (3.8)

    After fixing N1,N2,N6, and letting ϵ3=μ12N3, we then select ϵ1,ϵ2, and δ4 very small such that

    ρN2δ4>0,  μ1N6ϵ1>0,   μ4:=μ22N1ϵ2>0 (3.9)

    and select N3 large enough so that

    N3δIρ2N2C(1+1ϵ2)N6C(1+1ϵ1)>0. (3.10)

    Now, we note that αg2(s)h(s)=αg2(s)αg(s)g(s)<g(s); thus the dominated convergence theorem gives

    αCα=+0αg2(s)αg(s)g(s)ds0  as  α0. (3.11)

    Therefore, we can choose some 0<α0<1 such that for all 0<αα0,

    αCα<14C(1+N1(1+1ϵ1+1ϵ2)). (3.12)

    Finally, we select N so large enough and take α=1N So that

    λNN2CN3C(1+1ϵ3)N6C>0,N2CCα(1+N1(1+1ϵ1+1ϵ2))>0. (3.13)

    Combination of (3.6) - (3.13) yields the estimate (3.2). The equivalent relation (3.3) can be obtain easily by using Young's, Cauchy-Schwarz, and Poincaré's inequalities.

    Now, we state and prove our stability result for this subsection.

    Theorem 3.1. Assume Gρ=DIρ and (G1) and (G2) hold. Then, there exist positive constants a1 and a2 such that the energy solution (2.11) satisfies

    E(t)a2H11(a1tt0ξ(τ)dτ),   where  H1(t)=rt1τH(τ)dτ (3.14)

    and H1 is a strictly decreasing and strictly convex function on (0,r], with limt0H1(t)=+.

    Proof. Using the fact that g and ξ are positive, non-increasing and continuous, and H is positive and continuous, we have that for all t[0,t0]

    0<g(t0)g(t)g(0),  0<ξ(t0)ξ(t)ξ(0).

    Thus for some constants a,b>0, we obtain

    aξ(t)H(g(t))b.

    Therefore, for any t[0,t0], we get

    g(t)ξ(t)H(g(t))ag(0)g(0)ag(0)g(t) (3.15)

    and

    ξ(t)g(t)g(0)ag(t). (3.16)

    From (2.12) and (3.15), it follows that

    t00g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτg(0)at00g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτCE(t),  tt0. (3.17)

    From (3.2) and (3.17), we have

    L(t)βE(t)+12(g(3sxψx))(t)=βE(t)+12t00g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτ+12tt0g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτβE(t)CE(t)+12tt0g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτ.

    Thus, we get

    L1(t)βE(t)+12tt0g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτ, tt0, (3.18)

    where L1=L+CEE by virtue of (3.3). To finish our proof, we distinct two cases:

    Case 1: H(t) is linear. In this case, we multiply (3.18) by ξ(t), keeping in mind (2.12) and (G2), to get

    ξ(t)L1(t)βξ(t)E(t)+12ξ(t)tt0g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτβξ(t)E(t)+12tt0ξ(τ)g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτβξ(t)E(t)12tt0g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτβξ(t)E(t)CE(t),  tt0. (3.19)

    Therefore

    (ξL1+CE)(t)βξ(t)E(t),    tt0. (3.20)

    Since ξ is non-increasing and L1E, we have

    L2=ξL1+CEE. (3.21)

    Thus, from (3.20), we get for some positive constant α

    L2(t)βξ(t)E(t)αξ(t)L2(t),   tt0. (3.22)

    Integrating (3.22) over (t0,t) and recalling (3.21), we obtain

    E(t)a1ea2tt0ξ(s)ds=a1H11(a2tt0ξ(s)ds).

    Case 2: H(t) is nonlinear. In this case, we consider the functional L(t)=L(t)+I7(t). From (3.2) and Lemma 4.7 (see the Appendix), we obtain

    L(t)dE(t),  tt0, (3.23)

    where d>0 is a positive constant. Therefore,

    dtt0E(s)dsL(t0)L(t)L(t0).

    Hence, we get

    +0E(s)ds<. (3.24)

    Using (3.24), we define p(t) by

    p(t):=ηtt0(3sxψx)(t)(3sxψx)(tτ)22dτ,

    where 0<η<1 so that

    p(t)<1,tt0. (3.25)

    Moreover, we can assume p(t)>0 for all tt0; otherwise using (3.18), we obtain an exponential decay rate. We also define q(t) by

    q(t)=tt0g(τ)(3sxψx)(t)(3sxψx)(tτ)22dτ.

    Then q(t)CE(t), tt0. Now, we have that H is strictly convex on (0,r] (where r=g(t0)) and H(0)=0. Thus,

    H(στ)σH(τ),  0σ1 and τ(0,r]. (3.26)

    Using (3.26), condition (G2), (3.25), and Jensen's inequality, we get

    q(t)=1ηp(t)tt0p(t)(g(τ))η(3sxψx)(t)(3sxψx)(tτ)22dτ1ηp(t)tt0p(t)ξ(τ)H(g(τ))η(3sxψx)(t)(3sxψx)(tτ)22dτξ(t)ηp(t)tt0H(p(t)g(τ))η(3sxψx)(t)(3sxψx)(tτ)22dτξ(t)ηH(ηtt0g(τ)η(3sxψx)(t)(3sxψx)(tτ)22dτ)=ξ(t)ηˉH(ηtt0g(τ)η(3sxψx)(t)(3sxψx)(tτ)22dτ), (3.27)

    where ˉH is the convex extention of H on (0,+) (see remark 2.1). From (3.27), we have

    tt0g(τ)η(3sxψx)(t)(3sxψx)(tτ)22dτ1ηˉH1(ηq(t)ξ(t)).

    Therefore, (3.18) yields

    L1(t)βE(t)+CˉH1(ηq(t)ξ(t)),   tt0. (3.28)

    For r0<r, we define L3(t) by

    L3(t):=ˉH(r0E(t)E(0))L1(t)+E(t)E(t)

    since L1E. From (3.28) and using the fact that

    E(t)0, ˉH(t)>0, ˉH(t)>0,

    we obtain for all tt0

    L3(t)=r0E(t)E(0)ˉH(r0E(t)E(0))L1(t)+ˉH(r0E(t)E(0))L1(t)+E(t)βE(t)ˉH(r0E(t)E(0))+CˉH(r0E(t)E(0))ˉH1(ηq(t)ξ(t))+E(t). (3.29)

    Let us consider the convex conjugate of ˉH denoted by ˉH in the sense of Young (see [3] page 61-64). Thus,

    ˉH(τ)=τ(ˉH)1(τ)ˉH[(ˉH)(τ)] (3.30)

    and ˉH satisfies the generalized Young inequality

    ABˉH(A)+ˉH(B). (3.31)

    Let A=ˉH(r0E(t)E(0)) and B=ˉH1(μz(t)ξ(t)), It follows from (2.12) and (3.29)-(3.31) that

    L3(t)βE(t)ˉH(r0E(t)E(0))+CˉH(ˉH(r0E(t)E(0)))+Cηq(t)ξ(t)+E(t)βE(t)ˉH(r0E(t)E(0))+Cr0E(t)E(0)ˉH(r0E(t)E(0))+Cηq(t)ξ(t)+E(t). (3.32)

    Next, we multiply (3.32) by ξ(t) and recall that r0E(t)E(0)<r and

    ˉH(r0E(t)E(0))=H(r0E(t)E(0)),

    we arrive at

    ξ(t)L3(t)βξ(t)E(t)H(r0E(t)E(0))+Cr0E(t)E(0)ξ(t)H(r0E(t)E(0))+Cηq(t)+ξ(t)E(t)βξ(t)E(t)H(r0E(t)E(0))+Cr0E(t)E(0)ξ(t)H(r0E(t)E(0))CE(t). (3.33)

    Let L4(t)=ξ(t)L3(t)+CE(t). Since L3E, it follows that

    b0L4(t)E(t)b1L4(t), (3.34)

    for some b0,b1>0. Thus (3.33) gives

    L4(t)(βE(0)Cr0)ξ(t)E(t)E(0)ξ(t)H(r0E(t)E(0)), tt0.

    We select r0<r small enough so that βE(0)Cr0>0, we get

    L4(t)mξ(t)E(t)E(0)ξ(t)H(r0E(t)E(0))=mξ(t)H2(E(t)E(0)),  tt0, (3.35)

    for some constant m>0 and H2(τ)=τH(r0τ). We note here that

    H2(τ)=H(r0τ)+r0tH(r0τ),

    thus the strict convexity of H on (0,r], yields H2(τ)>0,H2(τ)>0 on (0,r]. Let

    F(t)=b0L4(t)E(0).

    From (3.34) and (3.35), we obtain

    F(t)E(t) (3.36)

    and

    F(t)=a0L4(t)(t)E(0)m1ξ(t)H2(F(t)), tt0. (3.37)

    Integrating (3.37) over (t0,t), we arrive at

    m1tt0ξ(τ)dτtt0F(τ)H2(F(τ))dτ=1r0r0F(t0)r0F(t)1τH(τ)dτ. (3.38)

    This implies

    F(t)1r0H11(¯m1tt0ξ(τ)dτ),   where  H1(t)=rt1τH(τ)dτ. (3.39)

    Using the properties of H, we see easily that H1 is strictly decreasing function on (0,r] and

    limt0H1(t)=+.

    Hence, (3.14) follows from (3.36) and (3.39). This completes the proof.

    Remark 3.1. The stability result in (3.1) is general and optimal in the sense that it agrees with the decay rate of g, see [10], Remark 2.3.

    Corollary 3.2. Suppose Gρ=DIρ, and (G1), and (G2) hold. Let the function H in (G2) be defined by

    H(τ)=τp,  1p<2, (3.40)

    then the solution energy (2.11) satisfies

    E(t)a2exp(a1t0ξ(τ)dτ),  for p=1,E(t)C(1+tt0ξ(τ)dτ)1p1,  for 1<p<2 (3.41)

    for some positive constants a2,a1 and C.

    In this subsection, we establish another stability result in the case non-equal speeds of wave propagation. To achieve this, we consider a stronger solution of (1.6). Let (w,3sψ,s,θ) be the strong solution of problem (1.6)–(1.8), then differentiation of 1.6 with respect to t gives

    {ρwttt+G(ψwx)xt=0,Iρ(3sψ)tttD(3sψ)xxt+t0g(τ)(3sψ)xxt(x,tτ)dτ+g(t)(3s0ψ0)xxG(ψwx)t=03Iρsttt3Dsxxt+3G(ψwx)t+4γst+δθxt=0,kθttλθxxt+δsxtt=0, (3.42)

    where (x,t)(0,1)×(0,+) and (3sψ)xx(x,0)=(3s0ψ0)xx. The modified energy functional associated to (3.42) is defined by

    E1(t)=12[ρwtt22+3Iρstt22+Iρ3sttψtt22+3Dsxt22+Gψtwxt22]+12[4γst22+kθt22+(Dt0g(τ)dτ)3sxtψxt22+(g(3sxtψxt))(t)]. (3.43)

    Lemma 3.2. Let (w,3sψ,s,θ) be the strong solution of problem (1.6)-(1.8). Then, the energy functional (3.43) satisfies, for all t0

    E1(t)=12(g(3sxtψxt))(t)12g(t)3sxtψxt22g(t)(3sttψtt),(3s0ψ0)xxλθxt22 (3.44)

    and

    E1(t)C(E1(0)+(3s0ψ0)xx22). (3.45)

    Proof. The proof of (3.44) follows the same steps as in the proof of Lemma 2.5. From (3.44), it is obvious that

    E1(t)g(t)(3sttψtt),(3s0ψ0)xx.

    So, using Cauchy-Schwarz inequality, we obtain

    E1(t)Iρg(t)23sttψtt22+g(t)2Iρ(3s0ψ0)xx22g(t)E1(t)+g(t)2Iρ(3s0ψ0)xx22. (3.46)

    This implies

    ddt(E1(t)et0g(τ)dτ)et0g(τ)dτg(t)2Iρ(3s0ψ0)xx22g(t)2Iρ(3s0ψ0)xx22 (3.47)

    Integrating (3.47) over (0,t) yields

    E1(t)e+0g(τ)dτE1(t)et0g(τ)dτE1(0)+12Iρ(t0g(τ)dτ)(3s0ψ0)xx22E1(0)+12Iρ(+0g(τ)dτ)(3s0ψ0)xx22. (3.48)

    Hence, (3.45) follows.

    Remark 3.2. Using Young's inequality, we observe from (3.44) and (3.45) that

    λθxt22=E1(t)+12(g(3sxtψxt))(t)12g(t)3sxtψxt22g(t)(3sttψtt),(3s0ψ0)xxE1(t)g(t)(3sttψtt),(3s0ψ0)xxE1(t)+g(t)(3sttψtt22+(3s0ψ0)xx22)E1(t)+g(t)(2IρE1(t)+(3s0ψ0)xx22)C(E1(t)+c1g(t)) (3.49)

    for some fixed positive constant c1. Similarly, we obtain

    0(g(3sxtψxt))(t)C(E1(t)+c1g(t)). (3.50)

    As in the case of equal-wave-speed of propagation, we define a Lyapunov functional

    ˜L(t)=˜NE(t)+6j=1~NjIj(t)+~N6I8(t), (3.51)

    where ˜N, ~Nj, j=1,2,3,4,5,6, are positive constants to be specified later and

    I8(t)=3λδ(IρGρD)10θxwxdx.

    Lemma 3.3. Suppose GρDIρ. Then, under suitable choice of ˜N, ~Nj, j=1,2,3,4,5,6, the Lyapunov functional ˜L satisfies, along the solution of (1.6), the estimate

    ˜L(t)˜βE(t)+12(g(3sxψx))(t)+C(E1(t)+c1g(t)), tt0, (3.52)

    for some positive constants ˜β and c1.

    Proof. Following the proof of Lemma 3.1, we end up with (3.52).

    Lemma 3.4. Suppose assumptions (G1) and (G2) hold and the function H in (G2) is linear. Let (w,3sψ,s,θ) be the strong solution of problem (1.6)-(1.8). Then,

    ξ(t)(g(3sxtψxt))(t)C(E1(t)+c1g(t)),  t0, (3.53)

    where c1 is a fixed positive constant.

    Proof. Using (3.50) and the fact that ξ is decreasing, we have

    ξ(t)(g(3sxtψxt))(t)=ξ(t)t0g(tτ)((3sxtψxt)(t)(3sxtψxt)(τ)22)dτt0ξ(tτ)g(tτ)((3sxtψxt)(t)(3sxtψxt)(τ)22)dτt0g(tτ)((3sxtψxt)(t)(3sxtψxt)(τ)22)dτ=(g(3sxtψxt))(t)C(E1(t)+c1g(t)). (3.54)

    Our stability result of this subsection is

    Theorem 3.3. Assume (G1) and (G2) hold and GρDIρ. Then, there exist positive constants a1,a2 and t2>t0 such that the energy solution (2.11) satisfies

    E(t)a2(tt0)H12(a1(tt0)tt2ξ(τ)dτ),t>t2,   where  H2(τ)=τH(τ). (3.55)

    Proof. Case 1: H is linear. Multiplying (3.52) by ξ(t) and using (G1), we get

    ξ(t)˜L(t)˜βξ(t)E(t)+12ξ(t)(g(3sxψx))(t)+Cξ(t)(E1(t)+c1g(t))˜βξ(t)E(t)CE(t)Cξ(0)E1(t)+ξ(0)c1g(t),  tt0

    Using the fact that \xi non-increasing, we obtain

    \begin{equation} \nonumber \left(\xi\tilde{L} + CE+ E_1 \right)'(t) \leq -\tilde{\beta} \xi(t) E(t) + c_2 g(t), \ \forall \ t\geq t_0. \end{equation}

    for some fixed positive constant c_2 . This implies

    \begin{equation} \tilde{\beta} \xi(t) E(t)\leq - \left(\xi\tilde{L} + CE+ E_1 \right)'(t) +c_2 g(t), \ \forall \ t\geq t_0. \end{equation} (3.56)

    Integrating (3.56) over (t_0, t) , using the fact that E is non-increasing and the inequality (3.45), we arrive at

    \begin{equation} \begin{aligned} \tilde{\beta} E(t)\int_{t_0}^t\xi(\tau)d\tau &\leq \tilde{\beta}\int_{t_0}^t \xi(\tau) E(\tau)d\tau\\ &\leq -\left(\xi\tilde{L} + CE+ E_1 \right)(t)+\left(\xi\tilde{L} + CE+ E_1 \right)(t_0)+ c_2\int_{t_0}^tg(\tau)d\tau\\ &\leq \left(\xi\tilde{L} + CE+ E_1 \right)(0)+ C\|(3s_0-\psi_0)_{xx}\|_2^2 + c_2\int_{0}^{\infty}g(\tau)d\tau\\ & = \left(\xi\tilde{L} + CE+ E_1 \right)(0)+ C\|(3s_0-\psi_0)_{xx}\|_2^2 +c_2(D-l_0). \end{aligned} \end{equation} (3.57)

    Thus, we have

    \begin{equation} E(t)\leq \frac{C}{ \int_{t_0}^t\xi(\tau)d\tau} , \ \forall \ t\geq t_0. \end{equation} (3.58)

    Case II: H is nonlinear. First, we observe from (3.52) that

    \begin{equation} \begin{aligned} \tilde{L}'(t)\leq &-\tilde{\beta} E(t)+ \frac{1}{2}\left( g\diamond (3s_x -\psi_x)\right)(t)+ C \left( -E'_1(t) + c_1g(t)\right)\\ \leq & -\tilde{\beta} E(t)+ C\left( \left( g\diamond (3s_x -\psi_x)\right)(t) +\left( g\diamond (3s_{xt} -\psi_{xt})\right)(t)\right)+ C \left( -E'_1(t) + c_1g(t)\right), \ \forall \ t\geq t_0. \end{aligned} \end{equation} (3.59)

    From (2.12), (3.16) and (3.50), we have for any t\geq t_0

    \begin{equation} \begin{aligned} \int_0^{t_0}g(\tau)\|(3s_x -&\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2d\tau+\int_0^{t_0}g(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2d\tau \\ & \quad \leq \frac{1}{\xi(t_0)}\int_0^{t_0}\xi(\tau)g(\tau)\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2d\tau\\ & \quad +\frac{1}{\xi(t_0)}\int_0^{t_0}\xi(\tau)g(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2d\tau\\ & \quad \leq - \frac{g(0)}{a\xi(t_0)}\int_0^{t_0}g'(\tau)\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2d\tau\\ & \quad -\frac{g(0)}{a\xi(t_0)}\int_0^{t_0}g'(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2d\tau\\ & \quad \leq-C\left(E'(t)+ E'_1(t) \right) +c_2 g(t), \end{aligned} \end{equation} (3.60)

    where c_2 is a fixed positive constant. Substituting (3.60) into (3.59), we obtain for any t\geq t_0

    \begin{equation} \begin{aligned} \tilde{L}'(t)\leq &-\tilde{\beta} E(t)-C\left(E'(t)+ E'_1(t) \right) + c_3g(t)+ C \int_{t_0}^{t}g(\tau) \|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau\\ & +C \int_{t_0}^{t}g(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau, \end{aligned} \end{equation} (3.61)

    where c_3 is a fixed positive constant. Now, we define the functional \Phi by

    \begin{equation} \begin{aligned} \Phi(t) = &\frac{\sigma}{t-t_0}\int_{t_0}^{t}\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau\\ &+\frac{\sigma}{t-t_0} \int_{t_0}^{t}\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau, \ \forall \ t \gt t_0. \end{aligned} \end{equation} (3.62)

    Using (2.11), (2.12), (3.43) and (3.45), we easily get

    \begin{equation} \begin{aligned} \frac{1}{t-t_0}\int_{t_0}^{t} \|(3s_x -\psi_x)(t)-&(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau+\frac{1}{t-t_0} \int_{t_0}^{t}\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau\\ &\leq \frac{2}{t-t_0}\int_{t_0}^{t}\left( \|(3s_x -\psi_x)(t)\|_2^2+\|(3s_x -\psi_x)(t-\tau)\|_2^2\right) d\tau\\ & \quad +\frac{2}{t-t_0}\int_{t_0}^{t}\left( \|(3s_{xt} -\psi_{xt})(t)\|_2^2+\|(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2\right) d\tau\\ &\leq \frac{4}{l_0(t-t_0)}\int_{t_0}^{t}\left(E(t)+E(t-\tau)+E_1(t)+ E_1(t-\tau) \right)d\tau \\ &\leq \frac{8}{l_0(t-t_0)}\int_{t_0}^{t}\left(E(0)+ C\left( E_1(0)+ \|(3s_0-\psi_0)_{xx}\|_2^2 \right) \right)d\tau \\ &\leq \frac{8}{l_0}\left(E(0)+ C\left( E_1(0)+ \|(3s_0-\psi_0)_{xx}\|_2^2\right) \right) \lt \infty, \ \forall \ t \gt t_0. \end{aligned} \end{equation} (3.63)

    This last inequality allows us to choose 0 < \sigma < 1 such that

    \begin{equation} \Phi(t) \lt 1, \ \ \forall \ t \gt t_0. \end{equation} (3.64)

    Hence forth, we assume \Phi(t) > 0 , otherwise, we get immediately from (3.61)

    \begin{equation} \nonumber E(t)\leq \frac{C}{t-t_0}, \ \ \forall \ t \gt t_0. \end{equation}

    Next, we define the functional \mu by

    \begin{equation} \begin{aligned} \mu(t) = -&\int_{t_0}^{t}g'(\tau)\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau\\ &-\int_{t_0}^{t}g'(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau \end{aligned} \end{equation} (3.65)

    and observe that

    \begin{equation} \mu(t)\leq -C\left(E'(t)+ E'_1(t) \right) + c_4g(t), \ \ \ \ \forall \ t \gt t_0, \end{equation} (3.66)

    where c_4 is a fixed positive constant. The fact that H is strictly convex and H(0) = 0 implies

    \begin{equation} H(\nu \tau)\leq \nu H(\tau), \ \ 0\leq \nu\leq 1 \ {\rm and}\ \tau\in (0, r]. \end{equation} (3.67)

    Using assumption (G1) , (3.67), Jensen’s inequality and (3.64), we get for any t > t_0

    \begin{equation} \begin{aligned} \mu(t) = &-\frac{1}{\Phi(t)}\int_{t_0}^{t}\Phi(t)g'(\tau)\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau\\ & \quad -\frac{1}{\Phi(t)}\int_{t_0}^{t}\Phi(t)g'(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau\\ &\geq \frac{1}{\Phi(t)}\int_{t_0}^{t}\Phi(t)\xi(\tau)H(g(\tau))\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau\\ & \quad +\frac{1}{\Phi(t)}\int_{t_0}^{t}\Phi(t)\xi(\tau)H(g(\tau))\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau\\ &\geq \frac{\xi(t)}{\Phi(t)}\int_{t_0}^{t}H(\Phi(t)g(\tau))\|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau\\ & \quad +\frac{\xi(t)}{\Phi(t)}\int_{t_0}^{t}H(\Phi(t)g(\tau))\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau\\ &\geq \frac{\xi(t)(t-t_0)}{\sigma}H\left(\frac{\sigma}{t-t_0} \int_{t_0}^{t}g(\tau)(\Omega_1(t-\tau)+\Omega_2(t-\tau)) d\tau\right) \\ & = \frac{\xi(t)(t-t_0)}{\sigma}\bar{H}\left(\frac{\sigma}{t-t_0} \int_{t_0}^{t}\left( \Omega_1(t-\tau)+\Omega_2(t-\tau)\right) d\tau\right), \end{aligned} \end{equation} (3.68)

    where

    \begin{equation} \nonumber \begin{aligned} &\Omega_1(t-\tau) = \|(3s_x -\psi_x)(t)-(3s_x -\psi_x)(t-\tau)\|_2^2 , \\ &\Omega_2(t-\tau) = \|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 \end{aligned} \end{equation}

    and \bar{H} is the C^2- strictly increasing and convex extension of H on (0, +\infty). This implies

    \begin{equation} \begin{aligned} \int_{t_0}^{t}g(\tau)\|(3s_x -\psi_x)(t)&-(3s_x -\psi_x)(t-\tau)\|_2^2 d\tau +\int_{t_0}^{t}g(\tau)\|(3s_{xt} -\psi_{xt})(t)-(3s_{xt} -\psi_{xt})(t-\tau)\|_2^2 d\tau\\ &\leq \frac{(t-t_0)}{\sigma}\bar{H}^{-1}\left(\frac{\sigma \mu(t)}{\xi(t)(t-t_0)} \right), \ \forall\ t \gt t_0. \end{aligned} \end{equation} (3.69)

    Thus, the inequality (3.61) becomes

    \begin{equation} \begin{aligned} \tilde{L}'(t)\leq &-\tilde{\beta} E(t)-C\left(E'(t)+ E'_1(t) \right) + c_3g(t)+ \frac{C(t-t_0)}{\sigma}\bar{H}^{-1}\left(\frac{\sigma \mu(t)}{\xi(t)(t-t_0)} \right), \ \forall\ t \gt t_0. \end{aligned} \end{equation} (3.70)

    Let \tilde{L}_1(t): = \tilde{L}(t)+C\left(E(t)+ E_1(t) \right). Then (3.70) becomes

    \begin{equation} \tilde{L}'_1(t)\leq -\tilde{\beta} E(t) + \frac{C(t-t_0)}{\sigma}\bar{H}^{-1}\left(\frac{\sigma \mu(t)}{\xi(t)(t-t_0)} \right) + c_3g(t), \ \forall\ t \gt t_0. \end{equation} (3.71)

    For 0 < r_1 < r, we define the functional \tilde{L}_2 by

    \begin{equation} \tilde{L}_2(t): = \bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\tilde{L}_1(t), , \ \forall\ t \gt t_0. \end{equation} (3.72)

    From (3.71) and the fact that

    E'(t)\leq 0, \ \bar{H}'(t) \gt 0, \ \bar{H}''(t) \gt 0,

    we obtain, for all t > t_0,

    \begin{align} \tilde{L}'_2(t) = &\left(-\frac{r_1}{(t-t_0)^2} .\frac{E(t)}{E(0)}+\frac{r_1}{t-t_0}.\frac{E'(t)}{E(0)}\right) \bar{H}''\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\tilde{L}_1(t)+\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\tilde{L}'_1(t)\\ \leq&-\tilde{\beta} E(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+c_3g(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+\frac{C(t-t_0)}{\sigma}\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\bar{H}^{-1}\left(\frac{\sigma \mu(t)}{\xi(t)(t-t_0)} \right). \end{align} (3.73)

    Let \bar{H}^{\ast} be the convex conjugate of \bar{H} as in (3.30) and let

    A = \bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\ \ {\rm and}\ \ B = \bar{H}^{-1}\left(\frac{\sigma \mu(t)}{\xi(t)(t-t_0)} \right).

    Then, (3.30), (3.31) and (3.73) yield, for all t > t_0,

    \begin{align} \tilde{L}'_2(t) \leq&-\tilde{\beta} E(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+c_3g(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\\ &+\frac{C(t-t_0)}{\sigma}\bar{H}^{\ast}\left(\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\right)+\frac{C(t-t_0)}{\sigma}.\frac{\sigma \mu(t)}{\xi(t)(t-t_0)}\\ \leq &-\tilde{\beta} E(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+c_3g(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+Cr_1\frac{E(t)}{E(0)}\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+C\frac{\mu(t)}{\xi(t)}\\ \leq&-(\tilde{\beta}E(0)-Cr_1)\frac{E(t)}{E(0)}\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+C\frac{\mu(t)}{\xi(t)}+c_3g(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right) \end{align} (3.74)

    By selecting r_1 small enough so that (\tilde{\beta}E(0)-Cr_1) > 0 , we arrive at

    \begin{equation} \begin{aligned} \tilde{L}'_2(t)\leq &-\tilde{\beta}_2\frac{E(t)}{E(0)}\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)+C\frac{\mu(t)}{\xi(t)}+c_3g(t)\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right), \ \forall\ t \gt t_0, \end{aligned} \end{equation} (3.75)

    for some positive constant \tilde{\beta}_2.

    Now, multiplying (3.75) by \xi(t) and recalling that r_1\frac{E(t)}{E(0)} < r , we arrive at

    \begin{align} \xi(t)\tilde{L}'_2(t)&\leq-\tilde{\beta}_2 \xi(t)\frac{E(t)}{E(0)}\bar{H}'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right) + C\mu(t)+c_3g(t)\xi(t)H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\\ &\leq -\tilde{\beta}_2 \xi(t)\frac{E(t)}{E(0)}H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)-C(E'(t)+E'_1(t))+c_4g(t)+c_3g(t)H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right), \ \forall\ t \gt t_0. \end{align} (3.76)

    Since \frac{r_1}{t-t_0}\longrightarrow 0 as t\longrightarrow \infty, there exists t_2 > t_0 such that \frac{r_1}{t-t_0} < r_1 , whenever t > t_2 . Using this fact and observing that H' strictly increasing, and E and \xi are non-decreasing, we get

    \begin{equation} H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\leq H'(r_1), \ \forall\ t \gt t_2. \end{equation} (3.77)

    Using (3.77), it follows from (3.76) that

    \begin{equation} \tilde{L}_3'(t)\leq-\tilde{\beta}_2 \xi(t)\frac{E(t)}{E(0)}H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right) + c_5g(t), \ \forall\ t \gt t_2, \end{equation} (3.78)

    where \tilde{L}_3 = (\xi\tilde{L}_2+CE+ CE_1) and c_5 > 0 is a constant. Using the non-increasing property of \xi, we have

    \begin{equation} \tilde{\beta}_2 \xi(t)\frac{E(t)}{E(0)}H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\leq -\tilde{L}_3'(t) + c_5g(t), \ \forall\ t \gt t_2. \end{equation} (3.79)

    Using the fact that E is non-increasing and H'' > 0 we conclude that the map

    \begin{equation} \nonumber t\longmapsto E(t)H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right) \end{equation}

    is non-increasing. Therefore, integrating (3.79) over (t_2, t) yields

    \begin{equation} \begin{aligned} \tilde{\beta}_2\frac{E(t)}{E(0)}H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\int_{t_2}^{t} \xi(\tau)d\tau &\leq \tilde{\beta}_2\int_{t_2}^{t} \xi(\tau)\frac{E(\tau)}{E(0)}H'\left( \frac{r_1}{\tau-t_0}.\frac{E(\tau)}{E(0)}\right)d\tau\\ &\leq -\tilde{L}_3(t)+\tilde{L}_3(t_2) + c_5\int_{t_2}^tg(\tau)d\tau\\ &\leq \tilde{L}_3(t_2) + c_5\int_{0}^{\infty} g(\tau)d\tau\\ & = \tilde{L}_3(t_2) + c_5(b-l_0), \ \forall\ t \gt t_2. \end{aligned} \end{equation} (3.80)

    Next, we multiply both sides of (3.80) by \frac{1}{t-t_0} , for t > t_2, we get

    \begin{equation} \frac{\tilde{\beta}_2}{(t-t_0)}.\frac{E(t)}{E(0)}H'\left( \frac{r_1}{t-t_0}.\frac{E(t)}{E(0)}\right)\int_{t_2}^{t} \xi(\tau)d\tau\leq \frac{\tilde{L}_3(t_2) + c_5(b-l_0)}{t-t_0}, \ \ \forall\ t \gt t_2. \end{equation} (3.81)

    Since H' is strictly increasing, then H_2(\tau) = \tau H'(\tau) is a strictly increasing function. It follows from (3.81) that

    \begin{equation} \nonumber E(t)\leq a_2(t-t_0) H_2^{-1} \left(\frac{a_1}{ (t-t_0)\int_{t_2}^t \xi (\tau)d\tau }\right), \ \forall\ t \gt t_2. \end{equation}

    for some positive constants a_1 and a_2. This completes the proof.

    (1). Let g(t) = ae^{-bt}, \ t\geq 0, \ \ a, \ b > 0 are constants and a is chosen such that ( G_1) holds. Then

    \begin{equation} \nonumber g'(t) = - abe^{-bt} = -bH(g(t)) \ \ {\rm with } \ \ H(t) = t. \end{equation}

    Therefore, from (3.14), the energy function (2.11) satisfies

    \begin{equation} E(t)\leq a_2e^{-\alpha t}, \ \forall \ t\geq 0, \ {\rm where}\ \ \alpha = ba_1. \end{equation} (3.82)

    Also, for H_2(\tau) = \tau , it follows from (3.55) that, there exists t_2 > 0 such that the energy function (2.11) satisfies

    \begin{equation} E(t)\leq \frac{C}{t-t_2}, \ \ \forall\ t \gt t_2, \end{equation} (3.83)

    for some positive constant C.

    (2). Let g(t) = ae^{-(1+t)^{b}}, \ t\geq 0, \ \ a > 0, \ 0 < b < 1 are constants and a is chosen such that ( G_1) holds. Then,

    \begin{equation} \nonumber g'(t) = -ab(1+t)^{b-1}e^{-(1+t)^{b}} = -\xi(t) H(g(t)), \end{equation}

    where \xi(t) = b(1+t)^{b-1} and H(t) = t. Thus, we get from (3.14) that

    \begin{equation} E(t)\leq a_2e^{-a_1(1+t)^b}, \ \forall \ t\geq 0. \end{equation} (3.84)

    Likewise, for H_2(t) = t , then estimate (3.55) implies there exists t_2 > 0 such that the energy function (2.11) satisfies

    \begin{equation} E(t)\leq \frac{C}{(1+t)^{b}}, \ \ \forall\ t \gt t_2, \end{equation} (3.85)

    for some positive constant C.

    (3). Let g(t) = \frac{a}{(1+t)^{b}}, \ t\geq 0, \ \ a > 0, \ b > 1 are constants and a is chosen in such a way that ( G_1) holds. We have

    \begin{equation} \nonumber g'(t) = \frac{-ab}{(1+t)^{b+1}} = -\xi\left(\frac{a}{(1+t)^b} \right)^{\frac{b+1}{b}} = -\xi g^q(t) = -\xi H(g(t)), \end{equation}

    where

    \begin{equation} \nonumber H(t) = t^q, \ \ q = \frac{b +1}{b} \ \ {\rm satisfying} \ \ 1 \lt q \lt 2 \ \ {\rm and }\ \ \xi = \frac{b}{a^\frac{1}{b}} \gt 0. \end{equation}

    Hence, we deduce from (3.41) that

    \begin{equation} E(t)\leq \frac{C}{(1+t)^b}, \ \forall \ t\geq 0. \end{equation} (3.86)

    Furthermore, for H_2(t) = qt^q , estimate (3.55) implies there exists t_2 > 0 such that the energy function (2.11) satisfies

    \begin{equation} E(t) \leq \frac{C}{(1+t)^{(b-1)/(b+1)}}, \ \forall\ t \gt t_2, \end{equation} (3.87)

    for some positive constant C.

    In this section, we prove the functionals L_i, i = 1\cdots8, used in the proof of our stability results.

    Lemma 4.1. The functional I_1(t) satisfies, along the solution of (1.6)-(1.8), for all t\geq t_0 > 0 and for any \epsilon_1, \epsilon_2 > 0 , the estimate

    \begin{align} I_1'(t)\leq -\frac{I_{\rho}g_0}{2} &\| 3s_t-\psi_t\|_2^2+\epsilon_1\|3s_x -\psi_x\|_2^2 + \epsilon_2\|\psi-w_x\|_2^2+ C C_{\alpha}\left( 1+\frac{1}{\epsilon_1}+\frac{1}{\epsilon_2}\right)\left(h\diamond (3s_x-\psi_x) \right)(t), \end{align} (4.1)

    where g_0 = \int_0^{t_0}g(\tau)d\tau\leq \int_0^{t}g(\tau)d\tau.

    Proof. Differentiating I_1(t) , using (1.6)_2 and integrating by part, we have

    \begin{align} I_1'(t) = - &I_{\rho}\int_0^1(3s_t-\psi_t) \int_0^t g'(t-\tau)\left((3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau dx\\ & + D(t) \int_0^1(3s_x-\psi_x)\int_0^t g(t-\tau)\left((3s_x-\psi_x)(t)- (3s_x-\psi_x)(\tau)\right)d\tau dx\\ & +\int_0^1 \left(\int_0^t g(t-\tau)\left( (3s_x-\psi_x)(t)- (3s_x-\psi_x)(\tau)\right)d\tau \right)^2dx-I_{\rho}\left( \int_0^t g(\tau)d\tau \right) \int_0^1(3s_t-\psi_t)^2 dx \\ &-G\int_0^1(\psi-w_x) \int_0^t g(t-\tau)\left(( 3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau dx, \end{align} (4.2)

    where D(t) = \left(D-\int_0^t g(\tau)d\tau\right). Now, we estimate the terms on the right hand-side of (4.2). Exploiting Young's and Poincaré's inequalities, Lemmas 2.1-2.6 and performing similar computations as in (2.8), we have for any \epsilon_1 > 0 ,

    \begin{align} D(t)\int_0^1(3s_x-\psi_x)&\int_0^t g(t-\tau)\left((3s_x-\psi_x)(t)- (3s_x-\psi_x)(\tau)\right)d\tau dx\\ &\leq \epsilon_1 \|(3s_x-\psi_x\|_2^2 + \frac{C C_{\alpha}}{\epsilon_1}\left(h\diamond (3s_x-\psi_x)\right)(t) \end{align} (4.3)

    and

    \begin{equation} \int_0^1 \left(\int_0^t g(t-\tau)\left( (3s_x-\psi_x)(t)- (3s_x-\psi_x)(\tau)\right)d\tau \right)^2dx\leq C_{\alpha}\left(h\diamond (3s_x-\psi_x)\right)(t). \end{equation} (4.4)

    Also, for \delta_1 > 0, we have

    \begin{align} - I_{\rho}&\int_0^1(3s_t-\psi_t) \int_0^t g'(t-\tau)\left((3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau dx\\ = &I_{\rho}\int_0^1(3s_t-\psi_t) \int_0^t h(t-\tau)\left((3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau dx\\ &-I_{\rho}\alpha\int_0^1(3s_t-\psi_t) \int_0^t g(t-\tau)\left((3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau dx\\ \leq&\delta_1\|3s_t-\psi_t\|_2^2 + \frac{I_{\rho}^2}{2\delta_1} \int_0^1 \left(\int_0^t h(t-\tau)\left( (3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau \right)^2dx\\ &+\frac{\alpha^2 I_{\rho}^2}{2\delta_1}\int_0^1 \left(\int_0^t g(t-\tau)\left( (3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau\right)^2dx\\ \leq& \delta_1\|3s_t-\psi_t\|_2^2 + \frac{ I_{\rho}^2}{2\delta_1}\left( \int_0^t h(\tau)d\tau\right) \left(h\diamond (3s-\psi)\right)(t)+\frac{\alpha^2 I_{\rho}^2 C_{\alpha}}{2\delta_1} \left(h\diamond (3s-\psi)\right)(t)\\ \leq& \delta_1\|3s_t-\psi_t\|_2^2 + \frac{C(C_{\alpha}+1)}{\delta_1} \left(h\diamond (3s-\psi)_x\right)(t). \end{align} (4.5)

    For the last term, we have

    \begin{align} -G\int_0^1(\psi-w_x)& \int_0^t g(t-\tau)\left(( 3s-\psi)(t)- (3s-\psi)(\tau)\right)d\tau dx\leq \epsilon_2\|\psi-w_x\|_2^2 + \frac{G^2 C_{\alpha}}{4\epsilon_2}\left(h\diamond (3s-\psi)_x\right)(t). \end{align} (4.6)

    Combination of (4.2)-(4.6) lead to

    \begin{align} I_1'(t)\leq -&\left( I_{\rho} \int_0^t g(\tau)d\tau-\delta_1\right) \|3w_t-\psi_t\|_2^2 +\epsilon_1\|3s_x -\psi_x\|_2^2+\epsilon_2\|\psi-w_x\|_2^2 \\ & + CC_{\alpha}\left( 1+\frac{1}{\delta_1}+\frac{1}{\epsilon_1}+\frac{1}{\epsilon_2}\right)\left(h\diamond (3s_x-\psi_x) \right)(t). \end{align} (4.7)

    Since g(0) > 0 and g is continuous. Thus for any t\geq t_0 > 0, we get

    \begin{equation} \int_0^{t}g(\tau)d\tau\geq \int_0^{t_0}g(\tau)d\tau = g_0 \gt 0. \end{equation} (4.8)

    We select \delta_1 = \dfrac{I_{\rho}g_0}{2} to get (4.1).

    Lemma 4.2. The functional I_2(t) satisfies, along the solution of (1.6)-(1.8) and for any \delta_4 > 0 , the estimate

    \begin{equation} \begin{aligned} I'_2(t)&\leq-3D\|s_x\|_2^2 -3\gamma \|s\|_2^2 + \delta_4\|w_t\|_2^2 + C\left(1+\frac{1}{\delta_4} \right) \|s_t\|_2^2 +C \|\theta_x\|_2^2, \ \ \forall t\geq 0. \end{aligned} \end{equation} (4.9)

    Proof. Differentiation of I_2(t) , using (1.6)_1 and (1.6)_3 and integration by part, leads to

    \begin{equation} \nonumber I_2'(t) = 3I_{\rho}\|s_t\|_2^2-3D\|s_x\|_2^2-4\gamma \|s\|_2^2-\delta\int_0^t s\theta_x dx + 3\rho\int_0^1w_t\int_0^x s_t (y)dy dx. \end{equation}

    Applying Cauchy-Schwarz and Young's inequalities and (2.5), we get for any \delta_4 > 0,

    \begin{eqnarray} I_2'(t)&\leq & 3I_{\rho}\|s_t\|_2^2-3D\|s_x\|_2^2-4\gamma \|s\|_2^2 + \gamma \|s\|_2^2 +\frac{\delta^2}{4\gamma} \|\theta_x\|_2^2 +\delta_4\|w_t\|_2^2 + \frac{9\rho^2}{4\delta_4}\int_0^1\left(\int_0^x s_t(y)dy \right)^2dx\\ &\leq& -3D\|s_x\|_2^2 -3\gamma \|s\|_2^2 + \delta_4\|w_t\|_2^2 + C\left(1+\frac{1}{\delta_4} \right) \|s_t\|_2^2 + C \|\theta_x\|_2^2. \end{eqnarray}

    This completes the proof.

    Lemma 4.3. The functional I_3(t) satisfies, along the solution of (1.6)-(1.8) and for any \epsilon_3 > 0, the estimate

    \begin{equation} \begin{aligned} I_3'(t)\leq& - \frac{\delta I_{\rho}}{2}\|s_t\|_2^2 + \epsilon_3 \|s_x\|_2^2 + \epsilon_3\|\psi-w_x\|_2^2 + C\left(1+\frac{1}{\epsilon_3} \right)\|\theta_x\|_2^2, \ \ \forall t\geq 0. \end{aligned} \end{equation} (4.10)

    Proof. Differentiation of I_3 , using (1.6)_3 , (1.6)_4 and integration by parts, yields

    \begin{align*} \nonumber I_3'(t) = 3&\lambda I_{\rho}\int_0^1 \theta_x s_t dx -3I_{\rho}\delta\|s_t\|_2^2 -3kD\int_0^1 \theta s_x dx +k\delta\|\theta\|_2^2\\ \nonumber &+ 3kG\int_0^1\theta\int_0^x (\psi-w_y)(y)dy dx +4\gamma k\int_0^1\theta\int_0^t s(y)dydx . \end{align*}

    Using Cauchy-Schwarz, Young's and Poincaré's inequalities together with Lemmas 2.1-2.6, we have

    \begin{eqnarray} I_3'(t)&\leq& \delta_2\|s_t\|_2^2 + C_{\delta_2}\|\theta_x\|_2^2 -3I_{\rho}\delta\|s_t\|_2^2 + \frac{\epsilon_3}{2}\|s_x\|_2^2 + C\left( 1+\frac{1}{\epsilon_3}\right) \|\theta\|_2^2 \\ &&+\epsilon_3\int_0^1\left(\int_0^x (\psi-w_y)(y)dy \right)^2 dx +\frac{\epsilon_3}{2}\int_0^1\left(\int_0^x s(y)dy \right)^2dx\\ &\leq & \delta_2\|s_t\|_2^2 + C_{\delta_2}\|\theta_x\|_2^2 -3I_{\rho}\delta\|s_t\|_2^2 + \epsilon_3\|s_x\|_2^2 +\epsilon_3\|\psi-w_x\|_2^2 +C\left( 1+\frac{1}{\epsilon_3}\right) \|\theta_x\|_2^2 . \end{eqnarray}

    We choose \delta_2 = \dfrac{5I_{\rho}\delta}{2} to get (4.10).

    Lemma 4.4. The functional I_4(t) satisfies, along the solution of (1.6)-(1.8) and for any \epsilon_4 > 0, the estimate

    \begin{equation} \begin{aligned} I_4'(t)\leq& -\rho\|w_t\|_2^2 + \epsilon_4\|3s_x-\psi_x\|_2^2+ C\|s_x\|_2^2 + C_{\epsilon_4} \|\psi-w_x\|_2^2 , \ \ \forall t\geq 0. \end{aligned} \end{equation} (4.11)

    Proof. Using (1.6)_1 and integration by parts, we have

    \begin{equation*} I_4'(t) = -\rho\|w_t\|_2^2- G\int_0^1(\psi-w_x) w_x dx. \end{equation*}

    We note that w_x = -(\psi-w_x)-(3s-\psi)+3s to arrive at

    \begin{equation} \nonumber I_4'(t) = -\rho\|w_t\|_2^2 + G\|\psi-w_x\|^2_2 + G\int_0^1(\psi-w_x)(3s-\psi)dx -3G\int_0^1(\psi-w_x)s dx. \end{equation}

    It follows from Young's and Poincaré's inequalities that

    \begin{align*} I_4'(t)&\leq -\rho\|w_t\|_2^2 + G\|\psi-w_x\|^2_2 + \epsilon_4\|3s-\psi\|_2^2 +\frac{C}{\epsilon_4}\|\psi-w_x\|^2_2+\frac{3G}{2} \|\psi-w_x\|^2_2 + \frac{3G}{2}\|s\|_2^2\\ \nonumber \leq&- \rho\|w_t\|_2^2 + G\|\psi-w_x\|^2_2 + \epsilon_4\|3s_x-\psi_x\|_2^2+ C \|s_x\|_2^2+C\left( 1+\frac{1}{\epsilon_4}\right)\|\psi-w_x\|^2_2. \end{align*}

    This completes the proof.

    Lemma 4.5. The functional I_5(t) satisfies, along the solution of (1.6)-(1.8) and for any 0 < \alpha < 1 , the estimate

    \begin{equation} I_5'(t)\leq -\frac{l_0}{4}\|3s_x-\psi_x\|_2^2 + I_{\rho} \|3s_t-\psi_t\|_2^2 + C\|\psi-w_x\|_2^2 + CC_{\alpha}\left(h\diamond (3s_x-\psi_x) \right)(t). \end{equation} (4.12)

    Proof. Differentiating I_5 , using (1.6)_2 , we arrive at

    \begin{eqnarray} I'_5(t)& = & I_{\rho} \|3s_t-\psi_t\|_2^2-\left(D-\int_0^t g(\tau)d\tau \right)\|3s_x-\psi_x\|_2^2+ G\int_0^1(3s-\psi)(\psi-w_x)dx \\ &&+\int_0^1(3s_x-\psi_x)\int_0^t g(t-\tau)\left(\left( 3s_x-\psi_x\right)(x, \tau)- \left(3s_x-\psi_x\right)(x, t)\right)d\tau dx. \end{eqnarray}

    Applying Lemmas 2.1-2.6, Cauchy-Schwarz, Young's and Poincaré's inequalities, we obtain any \delta_3 > 0

    \begin{eqnarray} I'_5(t)&\leq & I_{\rho} \|3s_t-\psi_t\|_2^2 -l_0\|3s_x-\psi_x\|_2^2 + \delta_3\|3s_x-\psi_x\|_2^2 +\frac{G^2}{4\delta_3}\|\psi-w_x\|_2^2\\ && +\frac{l_0}{2}\|3s_x-\psi_x\|_2^2 +\frac{1}{2l_0}C_{\alpha}\left(h\diamond (3s_x-\psi_x) \right)(t). \end{eqnarray} (4.13)

    We select \delta_3 = \dfrac{l_0}{4} and obtain the desired result.

    Lemma 4.6. The functional I_6(t) satisfies, along the solution of (1.6)-(1.8) and for any for any \epsilon_1 , the estimate

    \begin{align} I_6'(t)\leq-&G^2\|\psi-w_x\|_2^2 + \epsilon_1\|3s_t-\psi_t\|_2^2 + C\left(1+\frac{1}{\epsilon_1} \right)\|s_t\|_2^2\\ &+ C\|s_x\|^2_2 +C\|\theta_x\|_2^2+3(I_{\rho}G-\rho D)\int_0^1 w_ts_{xt} dx , \ \ \forall t\geq 0. \end{align} (4.14)

    Proof. Differentiating I_6(t) , using (1.6)_1 and (1.6)_3 and integration by parts, we obtain

    \begin{align} I'_6(t) = -&3G^2\|\psi-w_x\|_2^2-4\gamma G\int_0^1(\psi-w_x)s dx -\delta G \int_0^1 (\psi-w_x) \theta_x dx \\ &-3I_{\rho}G\int_0^t(3s_t-\psi_t)s_t dx + 9I_{\rho}G\|s_t\|_2^2 +3(I_{\rho}G-\rho D)\int_0^1 w_ts_{xt} dx. \end{align} (4.15)

    Young's and Poincaré's inequalities give

    \begin{eqnarray} &&-4\gamma G\int_0^1(\psi-w_x)s dx \leq G^2\|\psi-w_x\|_2^2 + 4\gamma^2 C_p\|s_x\|^2_2, \\ && -\delta G \int_0^1 (\psi-w_x) \theta_x dx\leq G^2\|\psi-w_x\|_2^2 +\frac{\delta^2}{4}\|\theta_x\|_2^2, \\ && -3I_{\rho}G\int_0^t(3s_t-\psi_t)s_t dx \leq \epsilon_1\|3s_t-\psi_t\|_2^2 +\frac{(3I_{\rho}G)^2}{\epsilon_1}\|s_t\|_2^2. \end{eqnarray} (4.16)

    Substituting (4.16) into (4.15), we obtain (4.14). This completes the proof.

    Lemma 4.7. The functional I_7(t) satisfies, along the solution of (1.6)-(1.8), the estimate

    \begin{equation} I_7'(t)\leq 3(D-l_0)\|3s_x-\psi_x\|_2^2- \frac{1}{2} (g\diamond (3s_x-\psi_x))(t), \ \forall t\geq 0. \end{equation} (4.17)

    Proof. Differentiate I_7(t) and use the fact that J'(t) = -g(t) to get

    \begin{equation} \begin{aligned} I'_7(t) = & \int_0^1 \int_0^t J'(t-\tau)(3s_x-\psi_x)^2(\tau)d\tau dx + J(0)\|3s_x-\psi_x\|_2^2\\ = & -(g\diamond (3s_x-\psi_x))(t) + J(t)\|3s_x-\psi_x\|_2^2\\ & \ -2\int_0^1 (3s_x-\psi_x)\int_0^t g(t-\tau)\left((3s_x-\psi_x)(\tau)-(3s_x-\psi_x)(t) \right)dx. \end{aligned} \end{equation} (4.18)

    Using Cauchy-Schwarz and (G1), we have

    \begin{equation} \begin{aligned} -2\int_0^1 &(3s_x-\psi_x)\int_0^t g(t-\tau)\left((3s_x-\psi_x)(\tau)-(3s_x-\psi_x)(t) \right)\\ \leq& 2(D-l_0)\|3s_x-\psi_x\|_2^2 +\frac{\int_0^t g(\tau)d\tau}{2(D-l_0)}(g\diamond (3s_x-\psi_x))(t)\\ \leq& 2(D-l_0)\|3s_x-\psi_x\|_2^2 +\frac{1}{2}(g\diamond (3s_x-\psi_x))(t) \end{aligned} \end{equation} (4.19)

    Thus, we get

    \begin{equation} I'_7(t)\le 2(D-l_0)\|3s_x-\psi_x\|_2^2- \frac{1}{2}(g\diamond (3s_x-\psi_x))(t) + J(t)\|3s_x-\psi_x\|_2^2. \end{equation} (4.20)

    Since J is decreasing (J'(t) = -g(t)\leq 0) , so J(t)\leq J(0) = D-l_0 . Hence, we arrive at

    \begin{equation} \nonumber I_7'(t)\leq 3(D-l_0)\|3s_x-\psi_x\|_2^2- \frac{1}{2} (g\diamond (3s_x-\psi_x))(t). \end{equation}

    The next lemma is used only in the proof of the stability result for nonequal-wave-speed of propagation.

    Lemma 4.8. Let (w, 3s-\psi, s, \theta) be the strong solution of problem (1.6). Then, for any positive numbers \sigma_1, \sigma_2, \sigma_3 , the functional I_8(t) satisfies

    \begin{equation} \begin{aligned} I'_8(t)\leq &-3(I_{\rho}G-\rho D)\int_0^1 w_ts_{xt} dx +\sigma_1\|w_t\|_2^2 + \sigma_2\|\psi-w_x\|_2^2 +\sigma_3\|3s_x-\psi_x\|_2^2\\ &+C\|s_x\|_2^2+C\left(1+\frac{1}{\sigma_1}+\frac{1}{\sigma_2}+\frac{1}{\sigma_3} \right) \|\theta_{xt}\|_2^2, \ \forall \ t\geq t_0. \end{aligned} \end{equation} (4.21)

    Proof. Differentiation of I_8 , using integration by part and the boundary condition give

    \begin{equation} \begin{aligned} I'_8(t)& = \frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_x w_{xt} dx+\frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt} w_x dx\\ & = \frac{3\lambda}{\delta}(I_{\rho} G-\rho D)\left[-\int_0^1\theta_{xx}w_t dx \right] +\frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt} w_x dx. \end{aligned} \end{equation} (4.22)

    We note that w_x = -(\psi-w_x)-(3s-\psi)+3s and from (1.6)_4, \ \lambda\theta_{xx} = k\theta_t +\delta s_{xt} . So, (4.22) becomes

    \begin{equation} \begin{aligned} I'_8(t) = &-\frac{3}{\delta}(I_{\rho}G-\rho D)k\int_0^1\theta_t w_{t} dx- 3(I_{\rho}G-\rho D)\int_0^1s_{xt} w_t dx+ \frac{9\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt}s dx\\ &-\frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt}(\psi-w_x) dx -\frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt} (3s-\psi)dx \end{aligned} \end{equation} (4.23)

    Using Young's and Poincaré's inequalities, we have for any positive numbers \sigma_1, \sigma_2, \sigma_3 ,

    \begin{equation} \begin{aligned} &-\frac{3}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_t w_{t} dx\leq \sigma_1\|w_t\|_2^2 +\frac{C}{\sigma_1} \|\theta_{xt}\|_2^2, \\ &-\frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt}(\psi-w_x) dx\leq \sigma_2\|\psi-w_x\|_2^2 +\frac{C}{\sigma_2} \|\theta_{xt}\|_2^2, \\ &-\frac{3\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt} (3s-\psi)dx\leq \sigma_3\|3s_x-\psi_x\|_2^2 + \frac{C}{\sigma_3} \|\theta_{xt}\|_2^2, \\ &\frac{9\lambda}{\delta}(I_{\rho}G-\rho D)\int_0^1\theta_{xt}s dx\leq C\|s_x\|_2^2 + C \|\theta_{xt}\|_2^2. \end{aligned} \end{equation} (4.24)

    Substituting (4.24) into (4.23), we obtain (4.21).

    In this paper, we have established a general and optimal stability estimates for a thermoelastic Laminated system, where the heat conduction is given by Fourier's Law and memory as the only source of damping. Our results are established under weaker conditions on the memory and physical parameters. From our results, we saw that the decay rate is faster provided the wave speeds of the first two equations of the system are equal (see (1.3)). A similar result was established recently in [19] when the heat conduction is given by Maxwell-Cattaneo's Law. An interesting case is when the kernel memory term is couple with the first or third equations in system (1.6). Our expectation is that the stability in both cases will depend on the speed of wave propagation.

    The authors appreciate the continuous support of University of Hafr Al Batin, KFUPM and University of Sharjah. The first and second authors are supported by University of Hafr Al Batin under project #G-106-2020 . The third author is sponsored by KFUPM under project #S B181018.

    The authors declare no conflict of interest



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