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Research article

On global existence for the stochastic nonlinear Schrödinger equation with time-dependent linear loss/gain

  • Received: 28 September 2024 Revised: 27 February 2025 Accepted: 04 June 2025 Published: 11 June 2025
  • This paper was focused on global existence for the stochastic nonlinear Schrödinger equation with time-dependent loss/gain, which read idu+(Δu+λ|u|αu+ia(t)u)dt=dW. We proved the global existence and uniqueness of the solution in H1(RN) through the uniform boundedness of the momentum and energy functionals. The global existence result of the solution for this type of equation depended on the ranges of time-dependent loss/gain coefficient.

    Citation: Lijun Miao, Jingwei Yu, Linlin Qiu. On global existence for the stochastic nonlinear Schrödinger equation with time-dependent linear loss/gain[J]. Electronic Research Archive, 2025, 33(6): 3571-3583. doi: 10.3934/era.2025159

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  • This paper was focused on global existence for the stochastic nonlinear Schrödinger equation with time-dependent loss/gain, which read idu+(Δu+λ|u|αu+ia(t)u)dt=dW. We proved the global existence and uniqueness of the solution in H1(RN) through the uniform boundedness of the momentum and energy functionals. The global existence result of the solution for this type of equation depended on the ranges of time-dependent loss/gain coefficient.



    The nonlinear Schrödinger equation with time-dependent coefficient, as one of the basic models for optics and Bose-Einstein condensates (BECs), has gained widespread attention in recent years (see [1,2] and references therein). It is logical to account for random effects disturbing the system. A standard approach in physics involves considering the Gaussian space-time white noise. Nevertheless, space-time white noise cannot be theoretically handled in mathematics, thus the noise is considered white in time and colored in space (see [3,4,5]).

    In this paper, we are concerned with the global existence of the solution for the following stochastic nonlinear Schrödinger equation with time-dependent linear loss/gain

    idu+(Δu+λ|u|αu+ia(t)u)dt=dW,(t,x)[0,)×RN (1.1)

    with

    u(0,x)=u0(x),xRN,

    where u0(x)H1(RN), λ=1, or λ=1 denotes that the nonlinearity is focusing or defocusing, 0<α<4N2 if N3 or α>0 if N=1,2, W denotes the complex valued Wiener process, a(t) is a real valued function defined on the interval [0,), and a(t)>0 or a(t)<0 describes the strength of loss or gain. For example, the time-dependent linear loss/gain term is described in the theory of BECs, where it represents the mechanism of continuously loading external atoms into the BECs (gain) by optical pumping or continuous depletion (loss) of atoms from the BECs (see [6]). From a phenomenological perspective, the time-dependent linear loss/gain term is used to explain the interaction of atomic clouds or thermal clouds (see [7]).

    Recently, the global existence for the stochastic nonlinear Schrödinger equation has been extensively studied and many important results have been achieved. For example, in [4], it is proved that the classical stochastic nonlinear Schrödinger equation with additive or multiplicative noise admits the global existence in H1(RN), respectively. In [5], it is showed that the defocusing energy-critical stochastic nonlinear Schrödinger equation with an additive noise admits the global existence by atomic spaces machinery and probabilistic perturbation argument. The authors in [8] study the global existence for the stochastic nonlinear Schrödinger equation with nonlinear Stratonovich noise in subcritical case, based on the deterministic and stochastic Strichartz's estimates. [9] investigates the defocusing mass critical nonlinear Schrödinger equation with a small multiplicative noise, it shows the global space-time bound by the decomposition of the solution. [10] proves that the solution of the stochastic nonlinear Schrödinger equation with linear multiplicative noise is global when defocusing, α=4N (mass-critical), N1 or α=4N2 (energy-critical), and N3 by rescaling transformation and the stability results. When a(t)=a>0, Equation (1.1) reduces to the weakly damped stochastic nonlinear Schrödinger equation. The global existence for this type of stochastic nonlinear Schrödinger equation driven by an additive noise is obtained (see [11,12]). [13] is devoted to the global existence for the stochastic nonlinear Schrödinger equation with weak damping driven by a multiplicative noise in mass-critical case. [14] uses the conservation of the L2(RN) norm and iteration argument to study the global existence for the random nonlinear Schrödinger equation with white noise dispersion and nonlinear time-dependent loss/gain in L2(RN) and H1(RN).

    Physically speaking, the nature of the function a(t) will have a significant impact on the behavior of the solution. [1] and [2] show the global existence result of the solution depending on the ranges of the time-dependent coefficient a(t). Inspired by the articles above, our main goal in this paper is to study the global existence of solution for Eq (1.1) with a(t) being time-dependent. Because of the loss of energy, the energy functional no longer satisfies the conservation law. In order to overcome the difficulty, setting u(t,x)=et0a(s)dsv(t,x) in Eq (1.1), v(t,x) satisfies

    idv+(Δv+λeαt0a(s)ds|v|αv)dt=et0a(s)dsdW,(t,x)[0,)×RN (1.2)

    with

    v(0,x)=u0(x),xRN.

    Therefore, in order to obtain the global existence of the solution for Eq (1.1), we only need to study the global existence of the solution for Eq (1.2). We use the uniform boundedness of the momentum and energy functionals to obtain the global existence for Eq (1.2). Our main theorem is as follows.

    Theorem 1.1. Let 0<α<4N2 if N3 or α>0 if N=1,2, and ϕL0,12 and v0 is an F0-measurable random variable with values in H1(RN). Assume that

    (1) either λ=1, 0<α<4N, a(t)L1(0,), a(t)<0,

    (2) or λ=1, a(t)L1loc(0,), a(t) permits sign-changing,

    then for every v0, there exists a unique global solution of Eq (1.2) in H1(RN), i.e., τ(v0)=+.

    According to Theoerm 1.1, we find that the global existence result of the solution depends on the ranges of the time-dependent loss/gain coefficient. In the absence of the time-dependent loss/gain term, i.e., a(t)=0, the conclusion of Theoerm 1.1 reduces to the well-established result presented in Theorem 3.4 in [4], thereby demonstrating the consistency of our result within the existing theoretical framework.

    This paper is organized as follows. In Section 2, we show the local existence for Eq (1.2) and study the evolution laws of the momentum and energy. In Section 3, under certain assumptions on λ, α, and a(t), we prove the global existence for Eq (1.2).

    In this section, we first introduce some mathematical spaces and estimates. Then, through using the method of [4], the local existence for Eq (1.2) is proved. Finally, we give the evolution laws of the momentum and energy.

    Throughout this paper, we use the following notations (see [15]). For p1, Lp(RN) denotes the Lebesgue space of p-integrable complex valued functions on RN. The Hilbert space L2(RN) is endowed with the norm and inner product

    ||u||L2(RN)=(RN|u(x)|2dx)12,
    (u,v)=ReRNu(x)ˉv(x)dx,u,vL2(RN).

    For sR, the Sobolev space Hs(RN) of tempered distribution uS(RN) whose Fourier transform ˆv satisfies (1+|ξ|2)s2ˆv(ξ)L2(RN). For a Banach space B, T>0 and p1, and Lp(0,T;B) denotes the space of functions from [0,T] into B with p-integrable over [0,T].

    Definition 2.1. (See [15,16]) The pair (r,q) is said to be admissible if 2r=N(121q) and 2q2NN2 when N3, or 2r when N=1,2.

    Lemma 2.2. (Strichartz's estimates). (See [15,16].) Let (r,q), (r1,q1), and (r2,q2) be admissible pairs; S(t)=eitΔ denotes the linear Schrödinger propagator, T>0, then the following properties hold,

    (i) for every gL2(RN), there exists a constant C such that

    ||S()g||Lr(0,T;Lq(RN))C||g||L2(RN),

    (ii) for every GLr2(0,T;Lq2(RN)), there exists a constant C such that

    T0S(ts)G(s)dsLr1(0,T;Lq1(RN))C||G||Lr2(0,T;Lq2(RN)),

    where r2 and q2 are the conjugates of r2 and q2.

    In order to state precisely Eqs (1.1) and (1.2), we consider the probability space (Ω,F,P,{Ft}t0). Let {βk}kN be a sequence of independent real valued Brownian motions associated to {Ft}t0, and let {ek}kN be an orthonormal basis of L2(RN). We consider the complex valued Wiener process

    W(t,x,ω)=kNβk(t,ω)ϕek(x),t0,xRN,ωΩ,

    where ϕL0,s2, which is the space of the Hilbert-Schmidt operator from L2(RN) into Hs(RN). The corresponding norm is then given by

    ϕ2L0,s2=tr(ϕϕ)=kNϕek2Hs(RN).

    Next, we study the local existence for Eq (1.2).

    Theorem 2.3. Let 0<α<4N2 if N3 or α>0 if N=1,2, a(t)L1(0,), ϕL0,12, and the initial data v0 is an F0-measurable random variable with values in H1(RN). Then, there exists a unique solution v to Eq (1.2) with continuous H1(RN) valued paths, such that v(0)=v0. This solution is defined on a random interval [0,τ(v0)), where τ(v0) is a stopping time such that

    τ(v0)=+orlimtτ(v0)||v(t)||H1(RN)=+.

    Proof. We use the mild form of Eq (1.2), that is,

    v(t)=S(t)v0+iλt0S(ts)(eαs0a(m)dm|v(s)|αv(s))dsit0S(ts)es0a(m)dmdW(s). (2.1)

    Set

    z(t)=it0S(ts)es0a(m)dmdW(s).

    By similar analysis to [4], we just need to prove zLr(0,T;W1,α+2(RN)) almost surely for any T>0, where (r,α+2) is an admissible pair and r=4(α+2)Nα. Since z is a Gaussian process and r>2, we have

    E(T0||z(s)||rLα+2(RN)ds)=T0E(||z(s)||rLα+2(RN))dsc1T0(E(||z(s)||α+2Lα+2(RN)))rα+2ds=c1T0(RNE(z|s,x|α+2)dx)rα+2dsc2T0(RN(E(|z(s,x)|2))α+22dx)rα+2ds. (2.2)

    Since

    E(|z(s,x)|2)=kNs0|S(sτ)eτ0a(m)dmϕek|2dτ,

    where {ek}kN is an orthonormal basis of L2(RN), by Minkowski's inequality and r>2, we deduce

    (RN(E(|z(s,x)|2))α+22dx)2α+2kNs0(RN|S(sτ)eτ0a(m)dmϕek|α+2dx)2α+2dτ=kN||S()eτ0a(m)dmϕek||2L2(0,s;Lα+2(RN))c3kN||S()eτ0a(m)dmϕek||2Lr(0,T;Lα+2(RN))c3kN||eτ0a(m)dmϕek||2L2(RN),

    where c3 depends only on r, α, and T, and Strichartz's estimates are used in the last inequality.

    Because a(t)L1(0,),

    (RN(E(|z(s,x)|2))α+22dx)2α+2c3eT0a(m)dmkN||ϕek||2L2(RN)=c4||ϕ||2L0,02, (2.3)

    where c4 depends only on r, α, T, and a(t). Combining (2.2) and (2.3), we get

    E(||z||rLr(0,T;Lα+2(RN)))c5||ϕ||2L0,02,

    where c5 depends only on r, α, T, and a(t). Since the spatial derivatives and S() commute, the same computation shows that

    E(||z()||rLr(0,T;W1,α+2(RN)))c6||ϕ||L0,12,

    which proves the Theorem 2.3.

    Remark 2.4. Suppose that u(t,x)=et0a(s)dsv(t,x), and we obtain the local existence for Eq (1.1).

    Now, we give the evolution laws of the momentum

    M(v)=v2L2(RN)

    and energy

    H(v,t)=12v2L2(RN)λα+2eαt0a(s)dsvα+2Lα+2(RN). (2.4)

    Proposition 2.5. Let α, a(t), ϕ, and v0 be as in Theorem 2.3. Then, for any stopping time τ such that τ<τ(v0) a.s., we have

    M(v(τ))=M(v0)2ImkNτ0RNves0a(m)dm¯ϕekdxdβk(s)+ϕ2L0,02τ0e2s0a(m)dmds. (2.5)

    Moreover, for any pR and p1, there exist constants Mp0, such that

    E(supt[0,τ]Mp(v(t)))MpE(Mp(v0)). (2.6)

    Proof. We apply the Itô formula given in [3] to M(v). Since M(v) is Fréchet derivable, the derivatives of M(v) along directions φ and (φ,ψ) are as follows,

    DM(v)(φ)=2ReRNv¯φdx,D2M(v)(φ,ψ)=2ReRNφ¯ψdx.

    Using the Itô formula yields

    dM(v(τ))=DM(v)(dv)+12D2M(v)(dv,dv). (2.7)

    For the first term of the righthand side of (2.7), we have

    DM(v)(dv)=2ReRNv¯dvdx=2ImkNRNves0a(m)dm¯ϕekdβk(s)dx.

    For the second term of the righthand side of (2.7), we have

    12D2M(v)(dv,dv)=ReRNdv¯dvdx=kNRNe2s0a(m)dm|ϕek|2dsdx.

    Integrating (2.7) over [0, τ], we get (2.5). We now prove (2.6). Applying the Itô formula to Mp(v) yields

    Mp(v(t))=Mp(v0)2pImkNt0Mp1(v)RNves0a(m)dm¯ϕek(x)dxdβk(s)+pϕ2L0,02t0e2s0a(m)dmMp1(v)ds+2p(p1)t0e2s0a(m)dmMp2(v)kNRe(RNv¯ϕek(x)dx)2ds.

    Taking the supremum and using a martingale inequality, it yields

    E(supt[0,τ]Mp(v(t)))E(Mp(v0))+6pE((τ0M2(p1)(v)ϕves0a(m)dm2L2(RN)ds)12)+pϕ2L0,02E(τ0e2s0a(m)dmMp1(v)ds)+2p(p1)E(τ0Mp2(v)ϕves0a(m)dm2L2(RN)ds)E(Mp(v0))+6p(τ0e2s0a(m)dmds)12ϕ2L0,02E(supt[0,τ]Mp12(v))+p(2p1)τ0e2s0a(m)dmdsϕ2L0,02E(supt[0,τ]Mp1(v)).

    By using Hölder's and Young's inequalities in the second term of the righthand side and an induction argument, (2.6) holds.

    Then, we give the evolution law of the energy.

    Proposition 2.6. Let α, a(t), ϕ, and v0 be as in Theorem 2.3. Then, for any stopping time τ such that τ<τ(v0) a.s., we have

    H(v,τ)=H(v0)+αλα+2τ0RNa(s)eαs0a(m)dm|v|α+2dxdsImRNτ0es0a(m)dm(Δ¯v+λeαs0a(m)dm|v|α¯v)dWdx+12kNτ0RNe2s0a(m)dm|ϕek|2dxdsλ2kNτ0RNe(2α)s0a(m)dm(|v|α|ϕek|2+α|v|α2(Im(ˉvϕek))2)dxds. (2.8)

    Proof. The proof is similar to Proposition 2.5. Since H(v,t) is Fréchet derivable, the derivatives of H(v,t) along directions φ and (φ,ψ) are as follows,

    DH(v,t)(φ)=ReRNv¯φdxλeαt0a(m)dmReRN|v|αv¯φdx,
    D2H(v,t)(φ,ψ)=ReRNψ¯φdxλeαt0a(m)dm(ReRN|v|αψ¯φdx+αRN|v|α2Re(v¯ψ)Re(v¯φ))dx.

    Using the Itô formula yields

    dH(v,t)=H(v,t)tdt+DH(v,t)(dv)+12D2H(v,t)(dv,dv). (2.9)

    For the first term of the righthand side of (2.9), we have

    H(v,t)t=αλα+2a(s)eαt0a(m)dmRN|v|α+2dx.

    For the second term of the righthand side of (2.9), we have

    DH(v,t)(dv)=ReRNv¯dvdxλeαt0a(m)dmReRN|v|αv¯dvdx=ImRNeαt0a(m)dmΔ¯vdWdxλe(1α)t0a(m)dmImRN|v|α¯vdWdx.

    For the last term of the righthand side of (2.9), we have

    12D2H(v,t)(dv,dv)=12(ReRNdv¯dvdxλeαt0a(m)dm(ReRN|v|αdv¯dvdx+αRN|v|α2Re(v¯dv)Re(v¯dv))dx)=12kNRNe2t0a(m)dm|ϕek|2dsdxλ2e(2α)t0a(m)dmkNRN|v|α|ϕek|2dsdxλα2e(2α)t0a(m)dmkNRN|v|α2(Im(¯vϕek))2dsdx.

    Integrating (2.9) over [0, τ], we get (2.8).

    In this section, our purpose is to prove the global existence for Eq (1.2), i.e., Theorem 1.1, via the uniform boundedness of the momentum and energy functionals. First, we have the following lemma.

    Lemma 3.1. Assume 0<α<4N2 if N3 or α>0 if N=1,2, and we have

    (1) λ=1, 0<α<4N, then

    v2L2(RN)83H(v,t)+C||v||2+4α4NαL2(RN),

    (2) λ=1, then

    v2L2(RN)2H(v,t).

    Proof. Case (2) is obvious, so we only need to prove case (1). When λ=1,

    H(v,t)=12v2L2(RN)1α+2eαt0a(s)dsvα+2Lα+2(RN).

    Using the Gagliardo-Nirenberg inequality and Young's inequality, the following estimation is obtained

    1α+2eαt0a(s)dsvα+2Lα+2(RN)C||v||α+2Nα2L2(RN)||v||Nα2L2(RN)18||v||2L2(RN)+C||v||2+4α4NαL2(RN). (3.1)

    Substituting (3.1) into H(v,t), we get

    12||v||2L2(RN)H(v,t)+18||v||2L2(RN)+C||v||2+4α4NαL2(RN).

    Then, case (1) holds. Next, we begin to estimate E(sup0tτv(t)2H1(RN)).

    Lemma 3.2. Let α, ϕ, and v0 be as in Theorem 2.3, and assume that

    (1) either λ=1, 0<α<4N, a(t)L1(0,), a(t)<0,

    (2) or λ=1, a(t)L1loc(0,), a(t) permits sign-changing,

    then for any given T0>0 and any stopping time τ with τ<inf(T0,τ(v0)) a.s., we have

    E(sup0tτv(t)2H1(RN))C(T0,ϕ,a(t),E(H(v0)),E(v02+4α4NαL2(RN))). (3.2)

    Proof. Supported by Proposition 2.5 and Lemma 3.1, we only need to prove the uniform boundedness of (2.8). Assume that v0L2+4α4Nα(Ω;L2(RN))L2(Ω;H1(RN)) and that E(H(v0)) is finite.

    Case (1): If λ=1, we neglect the last term in (2.8) since they are nonpositive. Taking the expectation and using a martingale inequality to (2.8), we have

    E(sup0tτH(v,t))E(H(v0))+αα+2E(τ0RN|a(s)|eαs0a(m)dm|v|α+2dxds)+3E((τ0es0a(m)dmϕ(Δˉv+eαs0a(m)dm|v|αˉv)2L2(RN)ds)12)+12ϕ2L0,12τ0e2s0a(m)dmds. (3.3)

    For the second term of the righthand side of (3.3), using Hölder's inequality, we have

    αα+2E(τ0RN|a(s)|eαs0a(m)dm|v|α+2dxds)αα+2E(τ0|a(s)|eαs0a(m)dmdssup0tτvα+2Lα+2(RN))αα+2T00|a(s)|eα0a(m)dmdsE(sup0tτvα+2Lα+2(RN)). (3.4)

    The validity of the last inequality in (3.4) depends critically on the condition a(t)<0. Using the Gagliardo-Nirenberg inequality and Young's inequality, we have

    αα+2T00|a(s)|eα0a(m)dmdsvα+2Lα+2(RN)C||v||α+2Nα2L2(RN)||v||Nα2L2(RN)18||v||2L2(RN)+C||v||2+4α4NαL2(RN). (3.5)

    Note that in the last inequality of (3.5), it is crucial that 0<α<4N. Substituting (3.5) into (3.4), and by Proposition 2.5, we get

    αα+2E(τ0RN|a(s)|eαs0a(m)dm|v|α+2dxds)C(E(||v0||2+4α4NαL2(RN)))+18E(sup0tτv2L2(RN)). (3.6)

    For the third term of the righthand side of (3.3), the operator ϕ is bounded from H1(RN) into L2(RN) with the norm majorized by ϕL0,12. Furthermore, H1(RN) is embedded into Lα+2(RN) and ϕ is also bounded from Lα+2α+1(RN) into L2(RN). We obtain,

    es0a(m)dmϕ(Δˉv+eαs0a(m)dm|v|αˉv)L2(RN)ϕL0,12es0a(m)dm(vL2(RN)+eαs0a(m)dmvα+1Lα+2(RN)).

    It follows that

    3E((τ0es0a(m)dmϕ(Δˉv+eαs0a(m)dm|v|αˉv)2L2(RN)ds)12)3ϕL0,12E((τ0e2s0a(m)dmv2L2(RN)ds)12)+3ϕL0,12E((τ0e(22α)s0a(m)dmv2α+2Lα+2(RN)ds)12). (3.7)

    For the first term of the righthand side of (3.7), using Hölder's and Young's inequalities, we have

    3ϕL0,12E((τ0e2s0a(m)dmv2L2(RN)ds)12)3ϕL0,12(T00e2s0a(m)dmds)12E(sup0tτvL2(RN))132E(sup0tτv2L2(RN))+C(T0,ϕ,a(t)). (3.8)

    For the second term of the righthand side of (3.7), using Hölder's inequality, Young's inequality and the Gagliardo-Nirenberg inequality, we have

    3ϕL0,12E((τ0e(22α)s0a(m)dmv2α+2Lα+2(RN)ds)12)3ϕL0,12(T00e(22α)s0a(m)dmds)12E(sup0tτvα+1Lα+2(RN))14(α+2)E(sup0tτvα+2Lα+2(RN))+C(T0,ϕ,a(t))132E(sup0tτv2L2(RN))+C(T0,ϕ,a(t),E(||v0||2+4α4NαL2(RN))). (3.9)

    Combining (3.7)–(3.9), we get

    3E((τ0es0a(m)dmϕ(Δˉv+eαs0a(m)dm|v|αˉv)2L2(RN)ds)12) 116E(sup0tτv2L2(RN))+C(T0,ϕ,a(t),E(||v0||2+4α4NαL2(RN))). (3.10)

    Therefore, together with Lemma 3.1, we finally obtain

    E(sup0tτH(v,t))E(H(v0))+C(T0,ϕ,a(t),E(||v0||2+4α4NαL2(RN)))+12E(sup0tτH(v,t)). (3.11)

    Then, case (1) holds.

    Case (2): If λ=1, taking the expectation and using a martingale inequality to (2.8), we have

    E(sup0tτH(v,t))E(H(v0))+αα+2E(τ0RN|a(s)|eαs0a(m)dm|v|α+2dxds)+3E((τ0es0a(m)dmϕ(Δˉv+eαs0a(m)dm|v|αˉv)2L2(RN)ds)12)+12ϕ2L0,12τ0e2s0a(m)dmds+12kNE(τ0RNe(2α)s0a(m)dm(|v|α|ϕek|2+α|v|α2(Im(ˉvϕek))2)dxds). (3.12)

    From (2.4), we obtain that

    1α+2eαt0a(s)dsvα+2Lα+2(RN)H(v,t). (3.13)

    Note that the condition a(t)L1loc(0,) ensures that the term eαt0a(s)ds in (3.13) remains nonnegative and bounded regardless of the sign of a(t) (positive or negative). It follows, for the second term of the righthand side of (3.12), that we have

    αα+2E(τ0RN|a(s)|eαs0a(m)dm|v|α+2dxds)αE(τ0|a(s)|H(v,s)ds)αE(τ0|a(s)|sup0sτH(v,s)ds). (3.14)

    For the third term of the righthand side of (3.12), using Hölder's inequality and Young's inequality, we have

    3E((τ0es0a(m)dmϕ(Δˉv+eαs0a(m)dm|v|αˉv)2L2(RN)ds)12)3ϕL0,12E((τ0e2s0a(m)dmv2L2(RN)ds)12)+3ϕL0,12E((τ0e(22α)s0a(m)dmv2α+2Lα+2(RN)ds)12)132E(sup0tτv2L2(RN))+C(T0,ϕ,a(t))+116(α+2)E(sup0tτeαt0a(m)dmvα+2Lα+2(RN))18E(sup0tτH(v,t))+C(T0,ϕ,a(t)). (3.15)

    For the last term in (3.12), using Hölder's inequality, we have

    12kNE(τ0RNe(2α)s0a(m)dm(|v|α|ϕek|2+α|v|α2(Im(ˉvϕek))2)dxds)1+α2kNE(τ0RNe(2α)s0a(m)dm|v|α|ϕek|2dxds)1+α2kNE(τ0e(2α)s0a(m)dmvαLα+2(RN)ϕek2Lα+2(RN)ds)18(α+2)E(sup0tτeαt0a(m)dmvα+2Lα+2(RN))+C(T0,ϕ,a(t))18E(sup0tτH(v,t))+C(T0,ϕ,a(t)). (3.16)

    Combining (3.14)–(3.16), and Gronwall's inequality, we finally have

    E(sup0tτH(v,t))C(T0,ϕ,a(t),E(H(v0))). (3.17)

    Then, case (2) holds. In conclusion, we finish the proof of Lemma 3.2.

    Remark 3.3. Suppose that u(t,x)=et0a(s)dsv(t,x), and we obtain the global existence for Eq (1.1).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by National Natural Science Foundation of China (No.12001256) and by the Fundamental Research Funds for the Universities of the Educational Department of Liaoning Provincial of China (No.JYTMS20231046).

    The authors declare there is no conflicts of interest.



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