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New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities

  • A system of three semilinear wave equations with strong external forces in Rn is considered. We use weighted phase spaces, where the problem is well defined, to compensate the lack of Poincare's inequality. Using the Faedo-Galerkin method and some energy estimates, we prove the existence of global solution. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincaré's inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in [16,29,31]. The work is relevant in the sense that the problem is more complex than what can be found in the literature.

    Citation: Yousif Altayeb. New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities[J]. AIMS Mathematics, 2021, 6(7): 7251-7265. doi: 10.3934/math.2021425

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  • A system of three semilinear wave equations with strong external forces in Rn is considered. We use weighted phase spaces, where the problem is well defined, to compensate the lack of Poincare's inequality. Using the Faedo-Galerkin method and some energy estimates, we prove the existence of global solution. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincaré's inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in [16,29,31]. The work is relevant in the sense that the problem is more complex than what can be found in the literature.



    We consider, for xRn, t>0, the following system

    {uttθΔ(u+t0ϖ1(ts)u(s)ds)+αut=h1(u,v,w)vttθΔ(v+t0ϖ2(ts)v(s)ds)+αvt=h2(u,v,w)wttθΔ(w+t0ϖ3(ts)w(s)ds)+αwt=h3(u,v,w)u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x)ut(x,0)=u1(x),vt(x,0)=v1(x),wt(x,0)=w1(x), (1.1)

    where n3,α>0, the functions hi(.,.,.)(R3,R),i=1,2,3 are given by

    h1(ξ1,ξ2,ξ3)=(q+1)[d|ξ1+ξ2+ξ3|(q1)(ξ1+ξ2+ξ3)+e|ξ1|(q3)/2ξ1|ξ2|(q+1)/2];
    h2(ξ1,ξ2,ξ3)=(q+1)[d|ξ1+ξ2+ξ3|(q1)(ξ1+ξ2+ξ3)+e|ξ2|(q3)/2ξ2|ξ3|(q+1)/2];
    h3(ξ1,ξ2,ξ3)=(q+1)[d|ξ1+ξ2+ξ3|(q1)(ξ1+ξ2+ξ3)+e|ξ3|(q3)/2ξ3|ξ1|(q+1)/2],

    with d,e>0,q>3. The function 1θ(x)ϑ(x)>0 for all xRn is a density such that

    ϑLτ(Rn)withτ=2n2nrn+2rfor2r2nn2. (1.2)

    As in [15], it is not hard to see that there exists a function GC1(R3,R) such that

    uh1(u,v,w)+vh2(u,v,w)+wh3(u,v,w)=(q+1)G(u,v,w), (u,v,w)R3. (1.3)

    satisfies

    (q+1)G(u,v,w)=|u+v+w|q+1+2|uv|(q+1)/2+2|vw|(q+1)/2+2|wu|(q+1)/2. (1.4)

    We define the function spaces H as the closure of C0(Rn), as in [18], we have

    H={vL2nn2(Rn)v(L2(Rn))n},

    with respect to the norm vH=(v,v)1/2H for the inner product

    (v,w)H=Rnvwdx,

    and L2ϑ(Rn) as that to the norm vL2ϑ=(v,v)1/2L2ϑ for

    (v,w)L2ϑ=Rnϑvwdx.

    For general r[1,+)

    vLrϑ=(Rnϑ|v|rdx)1r.

    is the norm of the weighted space Lrϑ(Rn).

    The main aim of this work is to consider an important problem from the point of view of application in sciences and engineering (materials which is something between that of elastic solids and Newtonian fluids), namely, a system of three wave equations having a damping effects in an unbounded domain with strong external forces including damping terms of memory type with past history. Using the Faedo-Galerkin [16] method and some energy estimates, we proved the existence of global solution in Rn owing to the weighted function. By imposing a new appropriate condition, with the help of some special estimates and generalized Poincaré's inequality, we obtained an unusual decay rate for the energy function. For more detail regarding the single equation, we review the following references [7,8]. The paper [7] is one of the pioneer in literature for the single equation, which is the source of inspiration of several researches, while the work [8] is a recent generalization of [7] by introducing less dissipative effects.

    To enrich our topic, it is necessary to review previous works regarding the nonlinear coupled system of wave equations, from a qualitative and quantitative study. Let us begin with the single wave equation treated in [13], where the aim goal was mainely on the system

    {utt+μutΔuωΔut=uln|u|, (x,t)Ω×(0,)u(x,t)=0,xΩ,t0u(x,0)=u0(x),ut(x,0)=u1(x),xΩ, (1.5)

    where Ω is a bounded domain of Rn, n1 with a smooth boundary Ω. The author firstly constructed a local existence of weak solution by using contraction mapping principle and of course showed the global existence, decay rate and infinite time blow up of the solution with condition on initial energy.

    Next, a nonexistence of global solutions for system of three semi-linear hyperbolic equations was introduced in [3]. A coupled system for semi-linear hyperbolic equations was investigated by many authors and a different results were obtained with the nonlinearities in the form f1=|u|q1|v|q+1u,f2=|v|q1|u|q+1v. (Please, see [2,5,9,14,24,29]).

    In the case of non-bounded domain Rn, we mention the paper recently published by T. Miyasita and Kh. Zennir in [16], where the considered equation as follows

    utt+autϕ(x)Δ(u+ωutt0g(ts)u(s)ds)=u|u|q1, (1.6)

    with initial data

    {u(x,0)=u0(x),ut(x,0)=u1(x). (1.7)

    The authors showed the existence of unique local solution and they continued to extend it to be global in time. The rate of the decay for solution was the main result by considering the relaxation function is strictly convex, for more results related to decay rate of solution of this type of problems, please see [6,17,25,26,30,31].

    Regarding the study of the coupled system of two nonlinear wave equations, it is worth recalling some of the work recently published. Baowei Feng et al. considered in [10], a coupled system for viscoelastic wave equations with nonlinear sources in bounded domain ((x,t)Ω×(0,)) with smooth boundary as follows

    {uttΔu+t0g(ts)Δu(s)ds+ut=f1(u,v)vttΔv+t0h(ts)Δv(s)ds+vt=f2(u,v). (1.8)

    Here, the authors concerned with a system in Rn(n=1,2,3). Under appropriate hypotheses, they established a general decay result by multiplication techniques to extends some existing results for a single equation to the case of a coupled system.

    It is worth noting here that there are several studies in this field and we particularly refer to the generalization that Shun et al. made in studying a complicate non-linear case with degenerate damping term in [22]. The IBVP for a system of nonlinear viscoelastic wave equations in a bounded domain was considered in the problem

    {uttΔu+t0g(ts)Δu(s)ds+(|u|k+|v|q)|ut|m1ut=f1(u,v) vttΔv+t0h(ts)Δv(s)ds+(|v|κ+|u|ρ)|vt|r1vt=f2(u,v)u(x,t)=v(x,t)=0,xΩ,t>0u(x,0)=u0(x),v(x,0)=v0(x)ut(x,0)=u1(x),vt(x,0)=v1(x), (1.9)

    where Ω is a bounded domain with a smooth boundary. Given certain conditions on the kernel functions, degenerate damping and nonlinear source terms, they got a decay rate of the energy function for some initial data.

    The lack of existence (Blow up) is considered one of the most important qualitative studies that must be spoken of, given its importance in terms of application in various applied sciences. Concerning the nonexistence of solution for a more degenerate case for coupled system of wave equations with different damping, we mention the papers [19,20,21,23,27].

    In m-equations, paper in [1] considered a system

    uitt+γuitΔui+ui=mi,j=1,ij|uj|pj|ui|piui, i=1,2,,m, (1.10)

    where the absence of global solutions with positive initial energy was investigated.

    We introduce a very useful Sobolev embedding and generalized Poincaré inequalities.

    Lemma 1.1. [16] Let ϑ satisfy (1.2). For positive constants Cτ>0 and CP>0 depending only on ϑ and n, we have

    v2nn2CτvH,

    and

    vL2ϑCPvH,

    for vH.

    Lemma 1.2. [12] Let ϑ satisfy (1.2), then the estimates

    vLrϑCrvH,

    and

    Cr=Cτϑ1rτ,

    hold for vH. Here τ=2n/(2nrn+2r) for 1r2n/(n2).

    We assume that the kernel functions ϖ1,ϖ2,ϖ3C1(R+,R+) satisfy

    {1¯ϖ1=l>0for¯ϖ1=+0ϖ1(s)ds, ϖ1(t)0,1¯ϖ2=m>0for¯ϖ2=+0ϖ2(s)ds, ϖ2(t)0,1¯ϖ3=ν>0for¯ϖ3=+0ϖ3(s)ds, ϖ3(t)0, (1.11)

    we mean by R+ the set {ττ0}. Noting by

    ϖ(t)=maxt0{ϖ1(t),ϖ2(t),ϖ3(t)}, (1.12)

    and

    ϖ0(t)=mint0{t0ϖ1(s)ds,t0ϖ2(s)ds,t0ϖ3(s)ds}. (1.13)

    We assume that there is a function χC1(R+,R+) such that

    ϖi(t)+χ(ϖi(t))0,χ(0)=0,χ(0)>0, i=1,2,3, (1.14)

    for any ξ0.

    Hölder and Young's inequalities give

    uv(q+1)/2L(q+1)/2ϑ(u2L(q+1)ϑ+v2L(q+1)ϑ)(q+1)/2(lu2H+mv2H)(q+1)/2, (1.15)

    and

    vw(q+1)/2L(q+1)/2ϑ(mv2H+νw2H)(q+1)/2, (1.16)

    and

    wu(q+1)/2L(q+1)/2ϑ(νw2H+lu2H)(q+1)/2. (1.17)

    Thanks to Minkowski's inequality to give

    u+v+w(q+1)L(q+1)ϑc(u2L(q+1)ϑ+v2L(q+1)ϑ+w2L(q+1)ϑ)(q+1)/2c(u2H+v2H+w2H)(q+1)/2.

    Then there exist η>0 such that

    u+v+w(q+1)L(q+1)ϑ+2uv(q+1)/2L(q+1)/2ϑ+2vw(q+1)/2L(q+1)/2ϑ+2wu(q+1)/2L(q+1)/2ϑη(lu2H+mv2H+νw2H)(q+1)/2. (1.18)

    We need to define positive constants λ0 and E0 by

    λ0η1/(q1)andE0=(121q+1)η2/(q1). (1.19)

    The mainely aim of the present paper is to obtain a novel decay rate of solution from the convexity property of the function χ given in Theorem 3.1.

    We denote as in [18,28] an eigenpair {(λi,ei)}iNR×H of

    θ(x)Δei=λieixRn,

    for any iN, (θ(x))1ϑ(x). Then

    0<λ1λ2λi+,

    holds and {ei} is a complete orthonormal system in H.

    Definition 1.3. The triplet functions (u,v,w) is said a weak solution to (1.1) on [0,T] if satisfies for xRn,

    {Rnϑ(x)(utt+αut)φdx+Rnuφdxt0ϖ1(ts)u(s)dsφdx=Rnϑ(x)h1(u,v,w)φdx,Rnϑ(x)(vtt+αvt)ψdx+Rnvψdxt0ϖ2(ts)v(s)dsψdx=Rnϑ(x)h2(u,v,w)ψdx,Rnϑ(x)(wtt+αwt)Ψdx+RnvΨdxt0ϖ3(ts)w(s)dsΨdx=Rnϑ(x)h3(u,v,w)Ψdx, (1.20)

    for all test functions φ,ψ,ΨH for almost all t[0,T].

    The next Theorem is concerned on the local solution (in time [0,T]).

    Theorem 2.1. (Local existence) Assume that

    1<qn+2n2andthatn3. (2.1)

    Let (u0,v0,w0)H3 and (u1,v1,w3)L2ϑ(Rn)×L2ϑ(Rn)×L2ϑ(Rn). Under the assumptions (1.2)–(1.17) and (1.11)–(1.14). Then (1.1) admits a unique local solution (u,v,w) such that

    (u,v,w)X3T, XTC([0,T];H)C1([0,T];L2ϑ(Rn)),

    for sufficiently small T>0.

    We prove the existence of global solution in time. Let us introduce the potential energy J:H3R defined by

    J(u,v,w)=(1t0ϖ1(s)ds)u2H+(ϖ1u)+(1t0ϖ2(s)ds)v2H+(ϖ2v)+(1t0ϖ3(s)ds)w2H+(ϖ3w), (2.2)

    where

    (ϖjw)(t)=t0ϖj(ts)w(t)w(s)2Hds,

    for any wL2(Rn),j=1,2,3. The modified energy is defined by

    E(t)=12(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+12J(u,v,w)Rnϑ(x)G(u,v,w)dx, (2.3)

    Theorem 2.2. (Global existence) Let (1.2)–(1.17) and (1.11)–(1.14) hold. Under (2.1) and for sufficiently small (u0,u1),(v0,v1),(w0,w1)H×L2ϑ(Rn), problem (1.1) admits a unique global solution (u,v,w) such that

    (u,v,w)X3, XC([0,+);H)C1([0,+);L2ϑ(Rn)). (2.4)

    The next, Lemma will play an important role in the sequel.

    Lemma 2.3. For (u,v,w)X3T, the functional E(t) associated with problem (1.1) is a decreasing energy.

    Proof. For 0t1<t2T, we have

    E(t2)E(t1)=t2t1ddtE(t)dt=12t2t1(ϖ1(t)u2H(ϖ1u))dt12t2t1(ϖ2(t)v2H(ϖ2v))dt12t2t1(ϖ3(t)w2H(ϖ3w))dtα(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)0,

    owing to (1.11)–(1.14).

    We sketch here the outline of the proof for local solution by a standard procedure(See [4,11,31]).

    Proof. (Of Theorem 2.1.) Let (u0,u1),(v0,v1),(w0,w1)H×L2ϑ(Rn). For any (u,v,w)X3T, we can obtain weak solution of the related system

    {ϑ(x)(ztt+αzt)Δz=t0ϖ1(ts)Δu(s)ds+ϑ(x)h1(u,v,w)ϑ(x)(ytt+αyt)Δy=t0ϖ2(ts)Δv(s)ds+ϑ(x)h2(u,v,w)ϑ(x)(ζtt+αζt)Δζ=t0ϖ3(ts)Δw(s)ds+ϑ(x)h3(u,v,w)z(x,0)=u0(x),y(x,0)=v0(x),ζ(x,0)=w0(x)zt(x,0)=u1(x),yt(x,0)=v1(x),ζt(x,0)=w1(x). (2.5)

    We reduces problem (2.5) to Cauchy problem for system of ODE by using the Faedo-Galerkin approximation. We then find a solution map :(u,v,w)(z,y,ζ) from X3T to X3T. We are now ready show that is a contraction mapping in an appropriate subset of X3T for a small T>0. Hence has a fixed point (u,v,w)=(u,v,w), which gives a unique solution in X3T.

    We will show the global solution. By using conditions on functions ϖ1,ϖ2,ϖ3, we have

    E(t)12J(u,v,w)Rnϑ(x)G(u,v,w)dx12J(u,v,w)1q+1u+v+w(q+1)L(q+1)ϑ2q+1(uv(q+1)/2L(q+1)/2ϑ+vw(q+1)/2L(q+1)/2ϑ+wu(q+1)/2L(q+1)/2ϑ)12J(u,v,w)ηq+1[lu2H+mv2H+νw2H](q+1)/212J(u,v,w)ηq+1(J(u,v,w))(q+1)/2=G(ς), (2.6)

    here ς2=J(u,v,w), for t[0,T), where

    G(ξ)=12ξ2ηq+1ξ(q+1).

    Noting that E0=G(λ0), given in (1.19). Then

    {G(ξ)0in  ξ[0,λ0]G(ξ)<0in  ξ>λ0. (2.7)

    Moreover, limξ+G(ξ). Then, we have the following lemma

    Lemma 2.4. Let 0E(0)<E0.

    (i) If u02H+v02H+w02H<λ20, then local solution of (1.1) satisfies

    J(u,v,w)<λ20, t[0,T).

    (ii) If u02H+v02H+w02H>λ20, then local solution of (1.1) satisfies

    u2H+v2H+w2H>λ21, t[0,T),λ1>λ0.

    Proof. Since 0E(0)<E0=G(λ0), there exist ξ1 and ξ2 such that G(ξ1)=G(ξ2)=E(0) with 0<ξ1<λ0<ξ2.

    The case (i). By (2.6), we have

    G(J(u0,v0,w0))E(0)=G(ξ1),

    which implies that J(u0,v0,w0)ξ21. Then we claim that J(u,v,w)ξ21, t[0,T). Moreover, there exists t0(0,T) such that

    ξ21<J(u(t0),v(t0),w(t0))<ξ22.

    Then

    G(J(u(t0),v(t0),w(t0)))>E(0)E(t0),

    by Lemma 2.3, which contradicts (2.6). Hence we have

    J(u,v,w)ξ21<λ20, t[0,T).

    The case (ii). We can now show that

    u02H+v02H+w02Hξ22,

    and

    u2H+v2H+w2Hξ22>λ20,

    in the same way as (i).

    Proof. (Of Theorem 2.2.) Let (u0,u1),(v0,v1),(w0,w1)H×L2ϑ(Rn) satisfy both 0E(0)<E0 and

    u02H+v02H+w02H<λ20.

    By Lemma 2.3 and Lemma 2.4, we have

    12(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+lu2H+mv2H+νw2H12(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+(1t0ϖ1(s)ds)u2H+(ϖ1u)+(1t0ϖ2(s)ds)u2H+(ϖ2v)+(1t0ϖ3(s)ds)w2H+(ϖ3w)2E(t)+2ηq+1[lu2H+mu2H+νw2H](q+1)/22E(0)+2ηq+1(J(u,v,w))(q+1)/22E0+2ηq+1λq+10=η2/(q1). (2.8)

    This completes the proof.

    Let

    Λ(u,v,w)=12(1t0ϖ1(s)ds)u2H+12(ϖ1u)+12(1t0ϖ2(s)ds)v2H+12(ϖ2v)+12(1t0ϖ3(s)ds)w2H+12(ϖ3w)Rnϑ(x)G(u,v,w)dx, (2.9)
    Π(u,v,w)=(1t0ϖ1(s)ds)u2H+(ϖ1u)+(1t0ϖ2(s)ds)v2H+(ϖ2v)+(1t0ϖ3(s)ds)w2H+(ϖ3w)(q+1)Rnϑ(x)G(u,v,w)dx. (2.10)

    Lemma 2.5. Let (u,v,w) be the solution of problem (1.1). If

    u02H+v02H+w02H(q+1)Rnϑ(x)G(u0,v0,w0)dx>0. (2.11)

    Then under condition (3.1), the functional Π(u,v,w)>0, t>0.

    Proof. By (2.11) and continuity, there exists a time t1>0 such that

    Π(u,v,w)0,t<t1.

    Let

    Y={(u,v,w)Π(u(t0),v(t0),w(t0))=0, Π(u,v,w)>0,t[0,t0)}. (2.12)

    Then, by (2.9), (2.10), we have for all (u,v,w)Y,

    Λ(u,v,w)=q12(q+1)[(1t0ϖ1(s)ds)u2H+(1t0ϖ2(s)ds)v2H+(1t0ϖ3(s)ds)w2H]+q12(q+1)[(ϖ1u)+(ϖ2v)+(ϖ3w)]+1q+1Π(u,v,w)q12(q+1)[lu2H+mv2H+νw2H+(ϖ1u)+(ϖ2v)+(ϖ3w)].

    Owing to (2.3), it follows for (u,v,w)Y

    lu2H+mv2H+νw2H2(q+1)q1Λ(u,v,w)2(q+1)q1E(t)2(q+1)q1E(0). (2.13)

    By (1.18), (3.1) we have

    (q+1)RnG(u(t0),v(t0),w(t0))η(lu(t0)2H+mv(t0)2H+νw(t0)2H)(q+1)/2η(2(q+1)q1E(0))(q1)/2(lu(t0)2H+mv(t0)2H+νw(t0)2H)γ(lu(t0)2H+mv(t0)2H+νw(t0)2H)<(1t00ϖ1(s)ds)u(t0)2H+(1t00ϖ2(s)ds)v(t0)2H+(1t00ϖ3(s)ds)w(t0)2H<(1t00ϖ1(s)ds)u(t0)2H+(1t00ϖ2(s)ds)v(t0)2H+(1t00ϖ3(s)ds)w(t0)2H+(ϖ1u)+(ϖ2v)+(ϖ3w), (2.14)

    hence Π(u(t0),v(t0),w(t0))>0 on Y, which contradicts the definition of Y since Π(u(t0),v(t0),w(t0))=0. Thus Π(u,v,w)>0, t>0.

    The decay rate for solution is given in the next Theorem

    Theorem 3.1. (Decay of solution) Let (1.2)–(1.17) and (1.11)–(1.14) hold. Under condition (2.1) and

    γ=η(2(q+1)q1E(0))(q1)/2<1, (3.1)

    there exists t0>0 depending only on ϖ1,ϖ2,ϖ3, λ1 and χ(0) such that

    0E(t)<E(t0)exp(tt0ϖ(s)1ϖ0(t)), (3.2)

    holds for all tt0.

    Proof. (Of Theorem 3.1.) By (1.18) and (2.13), we have for t0

    0<lu2H+mv2H+νw2H2(q+1)q1E(t). (3.3)

    Let

    I(t)=ϖ(t)1ϖ0(t),

    where ϖ and ϖ0 defined in (1.12) and (1.13).

    Noting that limt+ϖ(t)=0 by (1.11)–(1.13), we have

    limt+I(t)=0,  I(t)>0, t0.

    Then we take t0>0 such that

    0<12I(t)<χ(0),

    with (1.14) for all t>t0. Due to (2.3), we have

    E(t)12(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+12[(ϖ1u)+(ϖ2v)+(ϖ3w)]+12(1t0ϖ1(s)ds)u2H+12(1t0ϖ2(s)ds)v2H+12(1t0ϖ3(s)ds)w2H12(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+12[(ϖ1u)+(ϖ2v)+(ϖ3w)]+12(1ϖ0(t))[u2H+v2H+w2H].

    Then, by definition of I(t), we have

    I(t)E(t)12I(t)(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+12ϖ(t)[u2H+v2H+w2H]+12I(t)[(ϖ1u)+(ϖ2v)+(ϖ3w)], (3.4)

    and Lemma 2.3, we have for all t1,t20

    E(t2)E(t1)12t2t1(ϖ(t)[u2H+v2H+w2H])dt+12t2t1((ϖ1u)+(ϖ2v)+(ϖ3w))dtαt2t1(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)dt,

    then,

    E(t)12ϖ(t)[u2H+v2H+w2H]+12[(ϖ1u)+(ϖ2v)+(ϖ3w)]α(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ), (3.5)

    Finally, tt0, we have

    E(t)+I(t)E(t)(12I(t)α)(ut2L2ϑ+vt2L2ϑ+wt2L2ϑ)+12[(ϖ1u)+(ϖ2v)+(ϖ3w)]+12I(t)((ϖ1u)+(ϖ2v)+(ϖ3w)),

    and we can choose t0>0 large enough such that

    12I(t)<α,

    then

    E(t)+I(t)E(t)12t0{ϖ1(tτ)+I(t)ϖ2(tτ)}u(t)u(τ)2Hdτ+12t0{ϖ2(tτ)+I(t)ϖ2(tτ)}v(t)v(τ)2Hdτ+12t0{ϖ3(tτ)+I(t)ϖ3(tτ)}w(t)w(τ)2Hdτ12t0{ϖ1(τ)+I(t)ϖ1(τ)}u(t)u(tτ)2Hdτ+12t0{ϖ2(τ)+I(t)ϖ2(τ)}v(t)v(tτ)2Hdτ+12t0{ϖ3(τ)+I(t)ϖ3(τ)}w(t)w(tτ)2Hdτ12t0{χ(ϖ1(τ))+χ(0)ϖ1(τ)}u(t)u(tτ)2Hdτ+12t0{χ(ϖ2(τ))+χ(0)ϖ2(τ)}v(t)v(tτ)2Hdτ+12t0{χ(ϖ3(τ))+χ(0)ϖ3(τ)}w(t)w(tτ)2Hdτ0,

    by the convexity of χ and (1.14), we have

    χ(ξ)χ(0)+χ(0)ξ=χ(0)ξ.

    Then

    E(t)E(t0)exp(tt0I(s)ds),

    which completes the proof.

    The author would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

    The author agrees with the contents of the manuscript, and there is no conflict of interest among the author.



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