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Research article

Seismic response of utility tunnel systems embedded in a horizontal heterogeneous domain subjected to oblique incident SV-wave

  • A horizontal non-homogeneous field adversely affects the seismic resistance of both the utility tunnel and its internal pipes, with seismic waves obliquely incident on the underground structure causing more significant damages. To address these issues, this study, based on a viscous-spring artificial boundary, derives and validates the equivalent junction force formula for the horizontal non-homogeneous field. It then establishes a three-dimensional finite element model of the utility tunnel, pipes, and surrounding soil to obtain the acceleration and strain responses of the utility tunnel and its internal pipes under seismic loading. Finally, it investigates the impact of different incidence angles of shear waves (SV waves) on the response of the utility tunnel and its internal pipes. It was found that as the PGA increases from 0.1 to 0.4 g, both peak acceleration and strain of the utility tunnel and its internal pipes increase. The peak acceleration of the utility tunnel and pipes initially decreases and then increases with the angle of incidence, while the strain increases with the angle of incidence, reaching its peak value when the angle of incidence is 30°. The acceleration and strain responses of the utility tunnel and pipe are higher in sand than in clay, with the peak acceleration strongly correlating with the angle of incidence of ground shaking. The findings of this study provide valuable insights into the seismic design of horizontal non-homogeneous field utility tunnel systems.

    Citation: Hui-yue Wang, Sha-sha Yu, De-long Huang, Chang-lu Xu, Hang Cen, Qiang Liu, Zhong-ling Zong, Zi-Yuan Huang. Seismic response of utility tunnel systems embedded in a horizontal heterogeneous domain subjected to oblique incident SV-wave[J]. AIMS Geosciences, 2025, 11(1): 47-67. doi: 10.3934/geosci.2025004

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  • A horizontal non-homogeneous field adversely affects the seismic resistance of both the utility tunnel and its internal pipes, with seismic waves obliquely incident on the underground structure causing more significant damages. To address these issues, this study, based on a viscous-spring artificial boundary, derives and validates the equivalent junction force formula for the horizontal non-homogeneous field. It then establishes a three-dimensional finite element model of the utility tunnel, pipes, and surrounding soil to obtain the acceleration and strain responses of the utility tunnel and its internal pipes under seismic loading. Finally, it investigates the impact of different incidence angles of shear waves (SV waves) on the response of the utility tunnel and its internal pipes. It was found that as the PGA increases from 0.1 to 0.4 g, both peak acceleration and strain of the utility tunnel and its internal pipes increase. The peak acceleration of the utility tunnel and pipes initially decreases and then increases with the angle of incidence, while the strain increases with the angle of incidence, reaching its peak value when the angle of incidence is 30°. The acceleration and strain responses of the utility tunnel and pipe are higher in sand than in clay, with the peak acceleration strongly correlating with the angle of incidence of ground shaking. The findings of this study provide valuable insights into the seismic design of horizontal non-homogeneous field utility tunnel systems.



    In this paper, we study the positive solutions of the periodic-parabolic problem

    {ut=μk(x,t)Δu+m(x,t)uc(x,t)up, in Ω×R,uν=0, on Ω×R,u(x,0)=u(x,T), in Ω, (1.1)

    where Ω is a bounded domain of RN(N1) with smooth boundary Ω, ν is the outward normal vector of Ω, μ>0 and p>1 is constant, m(x,t)Cα,α2(ˉΩ×R)(0<α<1) is T-periodic in t, k(x,t), c(x,t)Cα,1(ˉΩ×R) are positive and T-periodic in t. It is known that the periodic reaction-diffusion equation (1.1) can be accurately used to describe different diffusion phenomena in infectious diseases, microbial growth, and population ecology, see [1,2,3,4]. From a biological point of view, Ω represents the habitat of species u and μk(x,t) stands for the diffusion rate, which is time and space dependent. The function m(x,t) represents the growth rate of species. In this situation, in the subset {(x,t)Ω×R:m(x,t)>0}, the species will increase, while in {(x,t)Ω×R:m(x,t)<0}, species will decrease. The coefficient c(x,t) means that environment Ω can accommodate species u. There are many interesting conclusions about the study of the reaction-diffusion equation, see [5,6,7,8] for the elliptic problems and [9,10,11,12,13,14] for the periodic problems.

    In particular, if k(x,t)k(t) for xˉΩ, problem (1.1) has been well investigated by Hess [2], Cantrell and Cosner [1]. Let λ(μ) be the unique principal eigenvalue of the eigenvalue problem

    {utμk(t)Δum(x,t)u=λ(μ)u, in Ω×R,uν=0, on Ω×R,u(x,0)=u(x,T), in Ω.

    It follows from [2,15] that Eq (1.1) has a positive periodic solution θμ(x,t) if and only if λ(μ)<0. In addition, Dancer and Hess [16] and Daners and López-Gómez [17] studied the effect of μ on the positive periodic solution of Eq (1.1) with various boundary conditions. The most interesting conclusion of [11,16,17] is that

    limμ0+θμ(x,t)=θ(x,t) locally uniformly  in Ω×[0,T],

    here θ(x,t) is the maximum nonnegative periodic solution of

    {ut=m(x,t)uc(x,t)up,tR,u(x,0)=u(x,T).

    However, there is little result on the associated large diffusion and the effect of large diffusion on positive solutions.

    Our goal is to study the existence and uniqueness of positive periodic solutions of Eq (1.1) and the asymptotic behavior of positive periodic solutions when the diffusion rate μ is large. To this end, let λ(μ;m) be the principal eigenvalue of

    {utμk(x,t)Δum(x,t)u=λ(μ;m)u, in Ω×R,uν=0, on Ω×R,u(x,0)=u(x,T), in Ω. (1.2)

    {It is well known that λ(μ;m) plays a major role in the study of the positive periodic solution of Eq (1.1). The properties of λ(μ;m) will be established in Section 2. In addition, let W2,pν(Ω)={uW2,p(Ω):uν=0}(N<p<). If u0W2,pν(Ω), it follows from [2] that the semilinear initial value problem}

    {ut=μk(x,t)Δu+m(x,t)uc(x,t)up, in Ω×R,uν=0, on Ω×R,u(x,0)=u0(x), in Ω,

    has a unique solution U(x,t)=U(x,t;u0) satisfying

    U(x,t)C1+α,1+α2(ˉΩ×[0,T])C2+α,1+α2(ˉΩ×(0,T]).

    Our first result is the existence and uniqueness of positive periodic solutions of Eq (1.1). For simplicity, in the rest of this paper, we use the following notations:

    k(t)=Ω1k(x,t)dx,m(t)=Ωm(x,t)k(x,t)dx,c(t)=Ωc(x,t)k(x,t)dx.

    Theorem 1.1. Suppose that T0m(t)dt>0. Then Eq (1.1) admits a unique positive periodic solution θμ(x,t) for all μ>0.

    Remark 1.1. By the result of Section 3, we know that there exists a unique positive solution to Eq (1.1) if and only if λ(μ;m)<0. In the case T0m(t)dt>0, we obtain that λ(μ;m)<0.

    Next, we study the asymptotic behavior of positive periodic solutions when the diffusion rate is large.

    Theorem 1.2. Suppose that

    m(t)>0 for t[0,T]. (1.3)

    Let θμ(x,t) be the unique positive periodic solution of Eq (1.1) for μ>0. Then we have

    limμθμ(x,t)=ω(t) in C1,12(ˉΩ×[0,T]), (1.4)

    where ω(t) is the unique positive periodic solution of

    {ut=m(t)k(t)uc(t)k(t)up,tR,u(0)=u(T). (1.5)

    Remark 1.2. With the approach of local upper-lower solutions developed by Daners and López-Gómez [17] in the study of classical periodic-parabolic logistic equations, we can prove that

    limμ0+θμ(x,t)=θ(x,t) locally uniformly  in Ω×[0,T],

    provided maxΩT0m(x,t)dt>0. It also shows that when m(t)<0<T0maxΩm(x,t)dt, populations with small dispersal rates survive, while populations with large dispersal rates perish. This means that a small diffusion rate is a better strategy than a large diffusion rate under appropriate circumstances.

    The rest of this paper is arranged as follows: In Section 2, we study the properties of principal eigenvalues for the periodic eigenvalue problems. In Section 3, we mainly study the existence, uniqueness and stability of the positive solution to Eq (1.1). Moreover, we investigate the asymptotic profiles of the positive periodic solution to Eq (1.1) as μ in Section 4.

    In this section, we consider the principal eigenvalue of Eq (1.2). To this end, we first study the linear initial value problem

    {utμk(x,t)Δua(x,t)u=0, in Ω×(τ,T],uν=0, on Ω×(τ,T],u(x,τ)=u0(x), in Ω, (2.1)

    where 0τ<T, u0W2,pν(Ω)(N<p<) and a(x,t)Cα,α2(ˉΩ×[τ,)). It is well known that there is a one-to-one correspondence between Eq (2.1) and the evolution operator Uμ(t,τ). Then we can define that u(x,t)=Uμ(t,τ)u0 is the solution of Eq (2.1). For simplicity, let X=Lp(Ω)(N<p<), X1=W2,pν(Ω) and

    F={uCα,α2(ˉΩ×R):u(,t+T)=u(,t) in R}.

    Inspired by the classical works of Hess [2], we first give some important results of Eq (2.1), which will be used in the rest of this paper.

    Lemma 2.1. If u0X is positive, then Uμ(t,τ)u0>0 in C1ν(ˉΩ) for 0τ<tT.

    Proof. Note that X1 is compactly embedded in X. The operator Uμ(t,τ)/X1:X1X1 can be continuously extended to the positive operator Uμ(t,τ)L(X,X1). Thus Uμ(t,τ)u00 in X1. Since sUμ(s,τ)u0 is continuous from [τ,T] to X1 and Uμ(τ,τ)u0=u00, we can get that Uμ(s,τ)u0>0 in X1 as s>τ goes to τ. In addition, we have

    Uμ(t,τ)u0=Uμ(t,s)Uμ(s,τ)u0,

    for τ<s<t. Thus it can be obtained that Uμ(t,τ)u0>0 for 0τ<tT.

    We now study the periodic-parabolic eigenvalue problem

    {utμk(x,t)Δua(x,t)u=λ(μ;a)u, in Ω×(0,T],uν=0, on Ω×(0,T],u(x,0)=u(x,T), in Ω. (2.2)

    If there is a nontrivial solution u(x,t) of Eq (2.2), then λ(μ;a) is called the eigenvalue. In particular, if u(x,t) is positive, then λ(μ;a) is the principal eigenvalue.

    Theorem 2.1. Let Kμ:=Uμ(T,0) and r be the spectral radius of Kμ. Then r is the principal eigenvalue of Kμ with positive eigenfunction u0 if and only if λ(μ;a)=1Tlnr is the principal eigenvalue of Eq (2.2) with positive eigenfunction u(x,t)=eλ(μ;a)tUμ(t,0)u0.

    Proof. It can be proved by the similar arguments as in [2, Proposition 14.4]. For the completeness, we provide a proof in the following. Suppose that r is the principal eigenvalue of Kμ with positive eigenfunction u0X1. Let u(x,t)=eλ(μ;a)tUμ(t,0)u0. Then u(x,t) satisfies

    {utμk(x,t)Δua(x,t)u=λ(μ;a)u, in Ω×(0,T],uν=0, on Ω×(0,T],u(x,0)=u0=1rKμu0=eλ(μ;a)TKμu0=u(x,T), in Ω.

    According to the regularity results, we have u(x,t)C2+α,1+α2(ˉΩ×R). This means that μ=1Tlnr is the principal eigenvalue of Eq (2.2), while u(x,t)=eλ(μ;a)tUμ(t,0)u0 is the corresponding positive eigenfunction.

    On the contrary, suppose that λ(μ;a)=1Tlnr is the eigenvalue of Eq (2.2) with positive eigenfunction u(x,t). Set v(x,t)=eλ(μ;a)tu(x,t). Then v(x,t) is the solution of

    {vtμk(x,t)Δva(x,t)v=0, in Ω×(0,T],vν=0, on Ω×(0,T],v(x,0)=u(x)=:u0, in Ω.

    Thus, we obtain v(x,t)=Uμ(t,0)u0 for 0tT and u0X1 is positive. Hence,

    v(T)=eλ(μ;a)Tu0=Kμu0.

    It follows from Krein-Rutman theorem that eλ(μ;a)T=r.

    Remark 2.1. For τ<t, it follows that Uμ(t,τ) is a compact and strongly positive operator on X1. Moreover, by Krein-Rutman theorem, we obtain r>0, and r is the unique principal eigenvalue of Kμ. This implies that Eq (2.2) has the unique principal eigenvalue λ(μ;a) for any μ>0.

    Lemma 2.2. Let a1(x,t), a(x,t)F satisfy

    a1(x,t)<a2(x,t) in ˉΩ×[0,T].

    Then λ(μ;a2)<λ(μ;a1) for any μ>0.

    Proof. Assume that there exists μ1>0 such that λ(μ1;a2)λ(μ1;a1). Let u1(x,t) and u2(x,t) be corresponding positive eigenfunctions, chosen in such a way that

    0<u1(x,t)<u2(x,t) in ˉΩ×[0,T].

    Then ω(x,t)=u2(x,t)u1(x,t) satisfies

    {ωtμ1k(x,t)Δωa1(x,t)ω>λ(μ1;a1)ω, in Ω×(0,T],ων=0, on Ω×(0,T],ω(x,0)=ω(x,T), in Ω.

    Set ϕ(x,t)=eλ(μ1;a1)tω(x,t), then we have

    {ϕtμ1k(x,t)Δϕa1(x,t)ϕ>0, in Ω×(0,T],ϕν=0, on Ω×(0,T],ϕ(x,0)=ω(x,0)=ω(x,T), in Ω.

    Thus, for any xΩ, we can obtain

    ϕ(x,T)>Kμ1ω(x,0) and ϕ(x,T)=eλ(μ1;a1)Tω(x,0).

    Hence, we obtain

    (eλ(μ1;a1)TKμ1)ω(x,0)>0 in X1.

    Note that ω(x,0)>0, it follows from [2, Theorem 7.3] that

    eλ(μ1;a1)T=rμ1<eλ(μ1;a1)T,

    where rμ1 is the principal eigenvalue of Kμ1. This is a contradiction.

    Lemma 2.3. Suppose that for any nN, an(x,t)F satisfies

    limnan(x,t)=a(x,t) in C1(ˉΩ×[0,T]).

    Then for fixed μ>0, we have

    limnλ(μ;an)=λ(μ;a).

    Proof. For any given ε>0, there exists nεN such that for any n>nε, there holds

    a(x,t)ε<an(x,t)<a(x,t)+ε in ˉΩ×[0,T].

    Notice that λ(μ;a±ε)=λ(μ;a)ε. From Lemma 2.2, we have

    λ(μ;a)ε<λ(μ;an)<λ(μ;a)+ε,

    for any n>nε.

    Lemma 2.4. Let λ(μ;a) be the principal eigenvalue of Eq (2.2) for μ>0. Then we have

    λ(μ;a)T0a(t)dtT0k(t)dt, (2.3)

    here a(t)=Ωa(x,t)k(x,t)dx.

    Proof. First, we consider the case

    T0Ωkt(x,t)k2(x,t)dxdt0.

    Let φ(x,t) be the positive eigenfunction corresponding to the principal eigenvalue λ(μ;a). Taking α>0 satisfies

    lnα=T0Ωkt(x,t)lnφ(x,t)k2(x,t)dxdtT0Ωkt(x,t)k2(x,t)dxdt.

    Then φα:=αφ(x,t) is also the principal eigenfunction of Eq (2.2). It is easy to obtain

    λ(μ;a)T0Ω1k(x,t)dxdt=T0Ωa(x,t)k(x,t)dxdtμT0ΩΔφαφαdxdt=T0Ωa(x,t)k(x,t)dxdtμT0Ω|Dφα|2φ2αdxdt. (2.4)

    This implies that Eq (3.2) holds.

    Next, we consider the case of

    T0Ωkt(x,t)k2(x,t)dxdt=0.

    We can find smooth T-periodic functions {kn(x,t)} such that

    limnkn(x,t)=k(x,t) in C(ˉΩ×[0,T]),

    and

    T0Ω(kn(x,t))tk2n(x,t)dxdt0.

    It follows from Lemma 2.3 that

    limnλn(μ;a)=λ(μ;a),

    where λn(μ;a) is the principal eigenvalue of Eq (2.2) with k(x,t) replaced by kn(x,t). It is clear from Eq (2.4) that

    λn(μ;a)=T0Ωa(x,t)kn(x,t)dxdtμT0Ω|Dφα|2φ2αdxdt.

    Letting n, we have Eq (3.2).

    Remark 2.2. In Eq (3.2), we obtain upper estimates for the principal eigenvalue of the Neumann problem Eq (2.2). Indeed, let λD be the principal eigenvalue of the eigenvalue problem

    {utμk(x,t)Δua(x,t)u=λDu, in Ω×(0,T],u=0, on Ω×(0,T],u(x,0)=u(x,T), in Ω.

    By a similar way as in [2], we can show

    λD1TT0[μk(x,s)+a(s)]ds,

    for any μ>0.

    In this section, we study the existence and uniqueness of positive solutions of Eq (1.1). First, we show that if Eq (1.2) has negative principal eigenvalues, then Eq (1.1) has a unique positive solution. To this end, we recall the upper-lower solutions of Eq (1.1). For the sake of convenience, let

    QT=Ω×(0,T],Q1=Ω×(0,T].

    Definition 3.1. The continuous function ˉu(x,t) is called the upper-solution of Eq (1.1), if

    {utμk(x,t)Δu+m(x,t)uc(x,t)up,inQT,uν0,onQ1,u(x,0)u(x,T),inΩ,

    is satisfied.

    The definition of the lower-solution is similar. We then can prove the following result, see [2,4,5,15].

    Theorem 3.1. Suppose that ˉu(x,t), u_(x,t) are a pair of ordered bounded upper-lower solutions of Eq (1.1). Then Eq (1.1) has a unique positive periodic solution θμ(x,t)C1+α,(1+α)/2(ˉQT) that satisfies

    u_(x,t)θμ(x,t)ˉu(x,t) in ˉQT.

    Proof. Let

    f(x,t,u)=m(x,t)uc(x,t)up.

    Then there exists a constant K>0 such that

    |f(x,t,u2)f(x,t,u1)|K|u2u1|,

    for any (x,t,ui)ˉQT×[u_(x,t),ˉu(x,t)], i=1,2. It follows from Lp theory that for any uCα,α2(ˉQT) satisfying [u_(x,t),ˉu(x,t)], the linear initial value problem

    {vtμk(x,t)Δv+Kv=Ku+f(x,t,u), in QT,vν=0, on Q1,v(x,0)=u(x,T), in Ω,

    admits a unique solution v. Thus, a nonlinear operator v=Fu is defined. We will prove that there is a fixed point for F in four steps.

    Step1. In this step, we prove that if u_u1u2ˉu, then u_v1=Fu1v2=Fu2ˉu.

    Take ω1=v2v1, then ω1 satisfies

    {[ω1]tμk(x,t)Δω1+Kω1=K(u2u1)+f(x,t,u2)f(x,t,u1)0, in QT,ω1ν=0, on Q1,ω1(x,0)=u2(x,T)u1(x,T)0, in Ω.

    By the comparison principle, we obtain ω10. This implies Fu2Fu1. Similarly, let ω2=v1u_, then ω2 satisfies

    {[ω2]tμk(x,t)Δω2+Kω2=K(u1u_)+f(x,t,u1)f(x,t,u_)0, in QT,ω2ν=0, on Q1,ω1(x,0)=u1(x,T)u_(x,T)0, in Ω.

    Thus, u_v1. Similarly, v2ˉu.

    Step2. In this step, we construct a convergent monotone sequence.

    The iterative sequences {un} and {vn} are constructed as follows:

    u1=Fˉu,u2=Fu1,,un=Fun1,
    v1=Fu_,v2=Fv1,,vn=Fvn1.

    Since u_ˉu and F is monotonically non-decreasing, then

    u_v1u1ˉu.

    Similarly, we obtain

    u_vnunˉu.

    And since u_u1ˉu,

    u_u2u1ˉu.

    By induction, we have un+1un. In the same way, we obtain vnvn+1. Thus, we have

    u_v1v2u2u1ˉu.

    This also implies that {un} and {vn} are monotonically bounded sequences, so there are u0(x,t) and v0(x,t) such that

    limnun(x,t)=u0(x,t),limnvn(x,t)=v0(x,t).

    Thus

    u_(x,t)v0(x,t)u0(x,t)ˉu(x,t) in ˉQT.

    Step3. In this step, we prove that u0(x,t) and v0(x,t) are solutions of Eq (1.1).

    Take E=W2,1p(QT)(p>n+2). First, we prove that F:DC(ˉQT) is a compact operator, where

    D={u(x,t)E:u_(x,t)u(x,t)ˉu(x,t) in ˉQT}.

    For u1, u2E, let v1=Fu1 and v2=Fu2, then ω3=v2v1 satisfies

    {[ω3]tμk(x,t)Δω3+Kω3=K(u2u1)+f(x,t,u2)f(x,t,u1), in QT,ω3ν=0, on Q1,ω1(x,0)=u2(x,T)u1(x,T), in Ω.

    By the Lp estimate and embedding theorem, it follows that

    ω3C1+α,1+α2(ˉQT)Cω3W2,1p(QT)C1(u2u1Lp(QT)+u2(x,T)u1(x,T)Lp(Ω)),

    here C and C1 are positive constants. Thus F:DC(ˉQT) is continuous. It is known from the embedding theorem that if u is bounded in W2,1p(QT), then Fu is bounded in C1+α,(1+α)/2(ˉQT). This means that F:DC(ˉQT) is a compact operator.

    Since un is bounded, un=Fun1 has a convergent subsequence in C(ˉQT). By the monotonicity of un,

    limnun(x,t)=u0(x,t) in C(ˉQT).

    Thus u0(x,t) is the solution of Eq (1.1) in W2,1p(QT). The embedding theorem is used again to get u0(x,t)C1+α,(1+α)/2(ˉQT). In the same way, we get that v0(x,t) is the classical solution of Eq (1.1).

    Step4. In this step, we prove the uniqueness and periodicity of the solution of Eq (1.1).

    Since k(x,t), m(x,t) and c(x,t) are periodic about t, then τ(x,t)=u0(x,t+T)u0(x,t) satisfies

    {τt(x,t)μk(x,t)Δτ(x,t)=m(x,t)τ(x,t)pc(x,t)˜up1(x,t)τ(x,t), in QT,τν=0, on Q1,τ(x,0)=0, in Ω, (3.1)

    here ˜u(x,t) is between u0(x,t+T) and u0(x,t). It is well known that the solution of Eq (3.1) is unique, thus u0(x,t+T)u0(x,t) in ˉQT.

    The uniqueness of the solution is based on the results of [2,4] and can also be found in recent research results [11,15]. Assume that v1 and v2 are two positive periodic solutions of Eq (1.1). We first prove that there exists a large constant M>1 such that

    M1v1<v2<Mv1 in QT.

    Indeed, it is clear that there exists M1>1 such that

    v2(x,0)=v2(x,T)<M1v1(x,T)=M1v1(x,0) in Ω.

    This implies v2(x,0) on \bar\Omega . Let \eta(x, t): = M_1v_1(x, t)-v_2(x, t) , then \eta(x, t) satisfies

    \begin{align*} \eta_t-\mu k(x,t)\Delta\eta = &m(x,t)\eta-c(x,t)[M_1v_1^p-v_2^p]\\ > &m(x,t)\eta-c(x,t)[(M_1v_1)^p-v_2^p]\\ = &m(x,t)\eta-pc(x,t)\varsigma^{p-1}(x,t)\eta, \end{align*}

    where \varsigma(x, t) is lying between M_1v_1(x, t) and v_2(x, t) . Notice that \frac{\partial\eta}{\partial\nu} = 0 on Q_1 . By the maximum principle, we have \eta > 0 in \bar Q_T . Similarly, we can obtain that there exists M_2 > 0 such that v_1 < M_2v_2 in \bar Q_T . Take M = \max\{M_1, M_2\} , then we have

    \begin{equation*} M^{-1}v_1 < v_2 < Mv_1\,\,\text{ in }\,\,Q_T. \end{equation*}

    We know that Mv_1(x, t) and M^{-1}v_1(x, t) are a pair of upper-lower solutions of Eq (1.1). According to the second step, Eq (1.1) has a minimum solution u_* and a maximum solution u^* , which satisfies u_*\leq v\leq u^* in \bar Q_T for all solution v satisfying M^{-1}v_1\leq v\leq Mv_1 . Thus, we obtain u_*\leq v_i\leq u^* for i = 1, 2 . Hence, u_*\equiv u^* implies the uniqueness of the solution to Eq (1.1). Set

    \begin{equation*} \alpha_{*} = \inf\left\{\alpha > 0\,|\,u^*(x,t)\leq \alpha u_*(x,t)\,\text{ in } \bar Q_T \right\}. \end{equation*}

    It is clear that \alpha_{*}\geq1 . Note that if \alpha_{*} = 1 , then u_*(x, t)\equiv u^*(x, t) in \bar Q_T . Assume that \alpha_{*} > 1 . Let \pi(x, t) = \alpha_{*} u_*-u^* . It is known from the maximum principle that \pi(x, t) > 0 in \bar Q_T . On the other hand, we know that

    \begin{equation*} \pi(x,0) = \pi(x,T)\geq\alpha_1 u_*(x,T) = \alpha_1 u_*(x,0)\,\,\text{ on }\,\,\bar\Omega, \end{equation*}

    for some small \alpha_1 > 0 . We can use the previous method to prove the existence of M to show that

    \begin{equation*} \pi(x,t)\geq\alpha_1 u_*(x,t)\,\,\text{ on }\,\,\bar Q_T. \end{equation*}

    Then we have u^*(x, t)\leq (\alpha_*-\alpha) u_*(x, t) . This is in contradiction with the definition of \alpha_* . Thus, we obtain \alpha_{*} = 1 . The uniqueness is proved.

    Lemma 3.1. If \lambda(\mu; m) < 0 , then Eq (1.1) admits a unique positive periodic solution \theta_\mu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) . Moreover, \theta_\mu(x, t) is globally asymptotically stable.

    Proof. Let \theta(x, t) be a principal eigenfunction of Eq (1.2) normalized by \|\theta(x, t)\|_{C(\bar Q_T)} = 1 . Then \underline{u} = \varepsilon\theta(x, t) is a lower-solution of Eq (1.1) for any

    \begin{equation*} 0 < \varepsilon\leq\left[\frac{-\lambda(\mu;m)}{\max_{\bar Q_T}c(x,t)}\right]^{1-p}. \end{equation*}

    Take

    \begin{equation*} M > \max\left\{1,\left[\frac{-\lambda(\mu;m)}{\min_{\bar Q_T}c(x,t)}\right]^{1-p}\right\}. \end{equation*}

    Then we have \bar u = M\theta(x, t) is an upper-solution of Eq (1.1). From Theorem 3.1, we get that Eq (1.1) has a unique positive solution \theta_\mu(x, t) .

    Since \theta_\mu(x, t) is the solution of Eq (1.1), then \lambda(\mu; m-c\theta_\mu^{p-1}) = 0 . Let \lambda_1 be the eigenvalue of the linear problem

    \begin{equation*} \begin{cases} u_t-\mu k(x,t)\Delta u-\left[m(x,t)-pc(x,t)\theta_\mu^{p-1}\right]u = \lambda_1u,&\text{ in } \Omega\times(0,T],\\ \frac{\partial u}{\partial\nu} = 0,&\text{ on } \partial\Omega\times(0,T],\\ u(x,0) = u(x,T),&\text{ in } \Omega. \end{cases} \end{equation*}

    Due to

    \begin{equation*} m(x,t)-c(x,t)\hat{u}^{p-1} > m(x,t)-pc(x,t)\hat{u}^{p-1}, \end{equation*}

    for u > 0 . Then \lambda_1 > 0 . It follows from Theorem 2.1 that r < 1 . Thus, \theta_\mu(x, t) is locally asymptotically stable. In addition, we can choose \varepsilon small enough and M large enough such that \varepsilon\theta(x, t) and M\theta(x, t) are a pair of ordered bounded upper-lower solutions of Eq (1.1). Then we know that \theta_\mu(x, t) is globally asymptotically stable by the standard iteration argument as in [2].

    Lemma 3.2. If (1.1) has a positive periodic solution, then \lambda(\mu; m) < 0 .

    Proof. Let \theta_\mu(x, t) be a positive periodic solution of Eq (1.1). Thanks to [2], we can get that Eq (1.1) is equivalent to

    \begin{equation*} (I-K_\mu)\theta_\mu(x,0) = -\int_0^TU_\mu(T,\tau)c(x,\tau)\theta_\mu^p(x,\tau)\,d\tau\,\,\text{ in }\,\,X_1. \end{equation*}

    Notice that \theta_\mu(x, t) > 0 . We now apply [2, Theorem 7.3] to obtain

    \begin{equation*} e^{-\lambda(\mu;m)T} > 1. \end{equation*}

    Thus, \lambda(\mu; m) < 0 .

    Proposition 3.1. If \int_0^Tm_*(t)\, dt > 0 , then Eq (1.1) admits a unique positive periodic solution \theta_\mu(x, t) for all \mu > 0 .

    Proof. According to Lemma 2.4, we know that

    \begin{equation} \lambda(\mu;m)\leq-\frac{\int_0^T m_*(t)\,dt}{\int_0^T k_*(t)\,dt}. \end{equation} (3.2)

    Due to \int_0^Tm_*(t)\, dt > 0 , \lambda(\mu; m) < 0 . This together with Lemma 3.1 implies that Eq (1.1) admits a unique positive periodic solution for all \mu > 0 .

    In this section, we study the asymptotic behavior of the positive periodic solution of Eq (1.1) when the diffusion rate is large. Here we use regularity estimates together with the perturbation technique to prove our main result. To do this, we first consider the perturbation equation

    \begin{equation} \begin{cases} u_t = \mu k(x,t)\Delta u+m(x,t)(u+\varepsilon)-c(x,t)u^{p},&\text{ in } Q_T,\\ \frac{\partial u}{\partial\nu} = 0,&\text{ on } Q_1,\\ u(x,0) = u(x,T),&\text{ in } \Omega, \end{cases} \end{equation} (4.1)

    where the parameter \varepsilon > 0 .

    Lemma 4.1. Assume that Eq (1.3) holds. Then Eq (4.1) has a positive periodic solution \theta^{\varepsilon}_{\mu}(x, t) for \mu > 0 , provided \varepsilon > 0 is small. Moreover, we can find \mu_1 > 0 such that

    \begin{equation} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \theta_\mu(x,t)\,\,\mathit{\text{ in }}\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} (4.2)

    for \mu\geq\mu_1 .

    Proof. Through a similar argument as in Theorem 3.1, we know the existence of the positive solution \theta^{\varepsilon}_{\mu}(x, t) to Eq (4.1). We only need to prove Eq (4.2). Let \sigma(x, t) = \theta^{\varepsilon}_{\mu}(x, \frac{t}{\mu}) . Then \sigma(x, t) satisfies

    \begin{equation*} \begin{cases} \sigma_t = k(x,\frac{t}{\mu})\Delta \sigma+\frac{1}{\mu}[m(x,\frac{t}{\mu})(\sigma+\varepsilon) -c(x,\frac{t}{\mu})\sigma^{p}],&\text{ in } Q_T,\\ \frac{\partial \sigma}{\partial\nu} = 0,&\text{ on } Q_1,\\ \sigma(x,0) = \theta^{\varepsilon}_{\mu}(x,0),&\text{ in } \Omega. \end{cases} \end{equation*}

    It is known from the L^p estimate that there exists \mu_1 > 0 such that \sigma(x, t) is bounded in W^{2, 1}_p(\Omega\times(0, \mu T]) for any \mu > \mu_1 . It then follows that \theta^{\varepsilon}_{\mu}(x, t) is bounded in W^{2, 1}_p(Q_T) for any \mu > \mu_1 . Then, taking p large enough, we know from the embedding theorem that \theta^{\varepsilon}_{\mu}(x, t) is compact in C^{1, \frac{1}{2}}(\bar Q_T) . Thus there is a subsequence, still denoted by \theta_\mu^{\varepsilon}(x, t) , such that

    \begin{equation} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \nu(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} (4.3)

    for some nonnegative periodic function \nu(x, t)\in C(\bar Q_T) . It follows from the argument of Lemma 3.1 that \varepsilon\theta(x, t) is a lower-solution of Eq (4.1). Thus we have \nu(x, t) > 0 for (x, t)\in\bar\Omega\times[0, T] . Since \theta^{\varepsilon}_{\mu}(x, t) is bounded and Eq (4.3), \nu satisfies

    \begin{equation*} \nu(x,t) = \nu(x,0)+\mu\int_0^t[k(x,s)\Delta\nu(x,s)+m(x,s)\nu -c(x,s)\nu^p]\,ds. \end{equation*}

    It is easy to obtain

    \begin{equation*} \begin{cases} \nu_t = \mu k(x,t)\Delta\nu+m(x,t)\nu-c(x,t)\nu^{p},&\text{ in } Q_T,\\ \frac{\partial\nu}{\partial\nu} = 0,&\text{ on } Q_1,\\ \nu(x,0) = \nu(x,T),&\text{ in } \Omega. \end{cases} \end{equation*}

    By standard parabolic regularity, we know that \nu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) . The uniqueness of the solution means that Eq (4.2) holds.

    At the end of this section, we prove Theorem 1.2.

    Proof of Theorem 1.2. We divide the proof into the following three steps.

    \bf{Step\; 1.} In this step, we prove that \theta_\mu(x, t) has a convergent subsequence as \mu\to\infty .

    It follows from a similar argument to Lemma 4.1 that there exists \mu_1 > 0 such that \theta_\mu(x, t) is compact in C^{1, \frac{1}{2}}(\bar Q_T) for any \mu > \mu_1 . Thus, by passing to a subsequence, there is a nonnegative periodic function \theta\in C(\bar Q_T) such that

    \begin{equation*} \lim\limits_{\mu\to\infty}\theta_\mu(x,t) = \theta(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T). \end{equation*}

    \bf{Step\; 2.} In this step, we show that \theta(x, t) is independent of t .

    Let f(t) be a smooth T -periodic function, then we have

    \begin{equation*} \begin{aligned} -\int_0^T\theta_\mu(x,t) f_t(t)\,dt = &\mu \int_0^T k(x,t)f(t)\Delta\theta_\mu(x,t)\,dt\\ &+\int_0^Tm(x,t)\theta_\mu(x,t)f(t)\,dt-\int_0^T c(x,t)\theta^{p}_\mu(x,t)f(t)\,dt. \end{aligned} \end{equation*}

    By dividing \mu and making \mu\to\infty , we have

    \begin{equation*} \int_0^T k(x,t)f(t)\Delta\theta(x,t)\,dt = 0. \end{equation*}

    Since f(t) is arbitrary, we obtain

    \begin{equation*} \Delta\theta(x,t) = 0. \end{equation*}

    Then we derive

    \begin{equation*} \int_{\Omega}|\nabla\theta(x,t)|^2\,dx = 0. \end{equation*}

    Thus we have \theta(x, t)\equiv\theta(t) for x\in\bar\Omega .

    \bf{Step\; 3.} In this step, we show that \theta(t) = \omega(t) in [0, T] .

    First, we assert that \theta(t)\in C^{1}((0, \infty)) . Indeed, it is easy to obtain from Eq (1.1) that

    \begin{equation*} \int_t^{t+\varepsilon}\int_{\Omega}\frac{u_s(x,s)}{k(x,s)}\,dxds = \int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}u(x,s)\,dxds -\int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}u^p(x,s)\,dxds. \end{equation*}

    Then we have

    \begin{equation*} \begin{aligned} &\int_{\Omega}\frac{u(x,t+\varepsilon)}{k(x,t+\varepsilon)}\,dx -\int_{\Omega}\frac{u(x,t)}{k(x,t)}\,dx+\int_t^{t+\varepsilon} \int_{\Omega}\frac{k_{t}(x,s)}{k^2(x,s)}u(x,s)\,dxds\\ = &\int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}u(x,s)\,dxds- \int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}u^p(x,s)\,dxds. \end{aligned} \end{equation*}

    Taking \mu\to\infty , we obtain

    \begin{equation*} \begin{aligned} &\int_{\Omega}\frac{\theta(t+\varepsilon)}{k(x,t+\varepsilon)}\,dx -\int_{\Omega}\frac{\theta(t)}{k(x,t)}\,dx+\int_t^{t+\varepsilon} \int_{\Omega}\frac{k_{t}(x,s)}{k^2(x,s)}\theta(s)\,dxds\\ = &\int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}\theta(s)\,dxds- \int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}\theta^p(s)\,dxds. \end{aligned} \end{equation*}

    Thus, we derive

    \begin{equation*} \left[\theta(t)\int_{\Omega}\frac{1}{k(x,t)}\,dx\right]_{t} = \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx\theta(t)-\int_{\Omega} \frac{c(x,t)}{k(x,t)}\,dx\theta^p(t)-\int_{\Omega}\frac{k_{t}(x,t)} {k^2(x,t)}\,dx\theta(t). \end{equation*}

    Hence,

    \begin{equation*} \begin{aligned} \theta_{t}(t) = &\frac{1}{k_*(t)}\int_{\Omega} \frac{m(x,t)}{k(x,t)}\,dx\theta(t) -\frac{1}{k_*(t)}\int_{\Omega}\frac{c(x,t)}{k(x,t)}\,dx\theta^p(t)\\ &-\frac{1}{k_*(t)}\int_{\Omega}\frac{k_{t}(x,t)} {k^2(x,t)}\,dx\theta(t)-\frac{[k_*(t)]_t}{k_*(t)}\theta(t), \end{aligned} \end{equation*}

    for t > 0 . Thus \theta(t)\in C^{1}((0, \infty)) holds.

    We then prove that \theta(t) satisfies Eq (1.5). It is obvious from Eq (1.1) that

    \begin{equation} \int_{\Omega}\frac{u_t(x,t)}{k(x,t)}\,dx = \int_{\Omega}\frac{m(x,t)}{k(x,t)}u(x,t)\,dx -\int_{\Omega}\frac{c(x,t)}{k(x,t)}u^p(x,t)\,dx. \end{equation} (4.4)

    Similarly, suppose that f(t) is a smooth T -periodic function. Multiplying f(t) on both sides of Eq (4.4) and integrating over [0, T] gives

    \begin{equation*} \begin{aligned} &-\int_0^T\int_{\Omega}u(x,t)\left[\frac{f(t)}{k(x,t)}\right]_{t}\,dxdt\\ = &\int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}u(x,t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}u^p(x,t)f(t)\,dxdt. \end{aligned} \end{equation*}

    Letting \mu\to\infty , we know

    \begin{equation*} -\int_0^T\int_{\Omega}\theta(t)\left[\frac{f(t)}{k(x,t)}\right]_{t}\,dxdt = \int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}\theta(t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}\theta^p(t)f(t)\,dxdt. \end{equation*}

    This implies

    \begin{equation*} \int_0^T\int_{\Omega}\frac{f(t)}{k(x,t)}\theta_{t}(t)\,dxdt = \int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}\theta(t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}\theta^p(t)f(t)\,dxdt. \end{equation*}

    By the arbitrary of f(t) , it follows that

    \begin{equation*} \int_{\Omega}\frac{1}{k(x,t)}\,dx\theta_{t}(t) = \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx\theta(t) -\int_{\Omega}\frac{c(x,t)}{k(x,t)}\,dx\theta^p(t). \end{equation*}

    Thus, we have

    \begin{equation*} \begin{cases} \theta_t = \frac{\tilde{M}(t)}{\tilde{k}(t)}\theta -\frac{\tilde{C}(t)}{\tilde{k}(t)}\theta^p,\quad t\in\mathbb{R},\\ \theta(0) = \theta(T). \end{cases} \end{equation*}

    Finally, we prove that \theta(t) > 0 in t\in[0, T] . We define \theta^{\varepsilon}_{\mu}(x, t) as the unique positive periodic solution of Eq (4.1) for small \varepsilon > 0 and large \mu . Similarly to the previous argument, it can be shown that

    \begin{equation} \lim\limits_{\mu\to\infty}\theta^{\varepsilon}_\mu(x,t) = \omega^{\varepsilon}(t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} (4.5)

    where \omega^{\varepsilon}(t) satisfies

    \begin{equation} \begin{cases} \omega^{\varepsilon}_t = \frac{M_*(t)}{k_*(t)} (\omega^{\varepsilon}+\varepsilon) -\frac{c_*(t)}{k_*(t)}(\omega^{\varepsilon})^p,\quad t\in\mathbb{R},\\ \omega^{\varepsilon}(0) = \omega^{\varepsilon}(T). \end{cases} \end{equation} (4.6)

    Since

    \begin{equation*} \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx > 0\,\,\text{ in }\,\,[0,T], \end{equation*}

    we can obtain that Eq (1.5) admits a unique periodic positive solution \omega(t) . Note that \omega(t) is a lower solution of Eq (4.6). Then there exists a unique positive periodic solution \omega^{\varepsilon}(t) to Eq (4.6). In addition, \omega^{\varepsilon}(t) is monotonically increasing about \varepsilon , and \omega_{\varepsilon}(t)\geq\omega(t) > 0 . We obtain that there exists a positive continuous function \omega_{0}(t) such that

    \begin{equation*} \lim\limits_{\varepsilon\to0+}\omega^{\varepsilon}(t) = \omega_{0}(t)\,\,\text{ in }\,\,[0,T]. \end{equation*}

    The uniqueness of the positive solution of Eq (4.6) implies that

    \begin{equation*} \lim\limits_{\varepsilon\to0+}\omega^{\varepsilon}(t) = \omega(t)\,\,\text{ in }\,\,[0,T]. \end{equation*}

    It follows from Lemma 4.1 that

    \begin{equation*} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \theta_\mu(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T). \end{equation*}

    This means that \theta(t) is positive, together with (4.4)–(4.6). Thus, we must have

    \begin{equation*} \theta(t)\equiv\omega(t)\,\,\text{ in }\,\,[0,T]. \end{equation*}

    This also implies that

    \begin{equation*} \lim\limits_{\mu\to\infty}\theta_\mu(x,t) = \omega(t) \,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation*}

    holds for the entire sequence.

    We consider the positive solutions of the periodic-parabolic logistic equation with indefinite weight function and nonhomogeneous diffusion coefficient. When the dispersal rate is small, we can obtain a similar result as in the homogeneous diffusion coefficient. Here we are interested in the case of large diffusion coefficient with nonhomogeneous diffusion coefficient.

    In Theorem 1.1, we obtain the condition of m(x, t) to guarantee a positive periodic solution for all \mu > 0 . Then we investigate the effect of large \mu on the positive solution and establish that the limiting profile is determined by the positive solution of Eq (1.5). More precisely, we prove that the positive periodic solution tends to the unique positive solution of the corresponding non-autonomous logistic equation when the diffusion rate is large.

    M. Fan was responsible for writing the original draft. J. Sun handled the review and supervision.

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

    The authors would like to thank the anonymous reviewers for the careful reading and several valuable comments to revise the paper. Fan was supported by Gansu postgraduate scientifc research (20230XZX-055) and Sun was supported by NSF of China (12371170) and NSF of Gansu Province of China (21JR7RA535, 21JR7RA537).

    The authors declare there is no conflict of interest.



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