In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.
Citation: Leqiang Zou, Yanzi Zhang. Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation[J]. Networks and Heterogeneous Media, 2025, 20(2): 387-405. doi: 10.3934/nhm.2025018
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In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.
In this article, we consider the following spatially discrete diffusion system with time delay
{∂tv1(x,t)=d1D[v1](x,t)−αv1(x,t)+h(v2(x,t−τ1)),∂tv2(x,t)=d2D[v2](x,t)−βv2(x,t)+g(v1(x,t−τ2)) | (1) |
with the initial data
vi(x,s)=vi0(x,s), x∈R, s∈[−τi,0], i=1,2, | (2) |
where
D[vi](x,t)=vi(x+1,t)−2vi(x,t)+vi(x−1,t), i=1,2. |
Here
According to (H1), the spatially homogeneous system of (1) admits two constant equilibria
(v1−,v2−)=0:=(0,0)and(v1+,v2+)=K:=(K1,K2). |
It is clear that (H1) is a basic assumption to ensure that system (1) is monostable on
{cϕ′1(ξ)=d1D[ϕ1](ξ)−αϕ1(ξ)+h(ϕ2(ξ−cτ1)),cϕ′2(ξ)=d2D[ϕ2](ξ)−βϕ2(ξ)+g(ϕ1(ξ−cτ2)),(ϕ1,ϕ2)(−∞)=(v1−,v2−),(ϕ1,ϕ2)(+∞)=(v1+,v2+), | (3) |
where
System (1) is a discrete version of classical epidemic model
{∂tv1(x,t)=d1∂xxv1(x,t)−a1v1(x,t)+h(v2(x,t−τ1)),∂tv2(x,t)=d2∂xxv2(x,t)−a2v2(x,t)+g(v1(x,t−τ2)). | (4) |
The existence and stability of traveling waves of (4) have been extensively studied, see [7,19,21,24] and references therein. Note that system (1) is also a delay version of the following system
{∂tv1(x,t)=d1D[v1](x,t)−a1v1(x,t)+h(v2(x,t)),∂tv2(x,t)=d2D[v2](x,t)−a2v2(x,t)+g(v1(x,t)). | (5) |
When system (5) is a quasi-monotone system, Yu, Wan and Hsu [27] established the existence and stability of traveling waves of (5). To the best of our knowledge, when systems (1) and (5) are non-quasi-monotone systems, no result on the existence and stability of traveling waves has been reported. We should point out that the existence of traveling waves of (1) can be easily obtained. Hence, the main purpose of the current paper is to establish the stability of traveling waves of (1).
The stability of traveling waves for the classical reaction-diffusion equations with and without time delay has been extensively investigated, see e.g., [4,9,10,12,13,14,16,22,24]. Compared to the rich results for the classical reaction-diffusion equations, limited results exist for the spatial discrete diffusion equations. Chen and Guo [1] took the squeezing technique to prove the asymptotic stability of traveling waves for discrete quasilinear monostable equations without time delay. Guo and Zimmer [5] proved the global stability of traveling wavefronts for spatially discrete equations with nonlocal delay effects by using a combination of the weighted energy method and the Green function technique. Tian and Zhang [19] investigated the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with two species by the weighted energy method together with the comparison principle. Later on, Chen, Wu and Hsu [2] employed the similar method to show the global stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with three species. We should point out that the methods for the above stability results heavily depend on the monotonicity of equations and the comparison principle. However, the most interesting cases are the equations without monotonicity. It is known that when the evolution equations are non-monotone, the comparison principle is not applicable. Thus, the methods, such as the squeezing technique, the weighted energy method combining with the comparison principle are not valid for the stability of traveling waves of the spatial discrete diffusion equations without monotonicity.
Recently, the technical weighted energy method without the comparison principle has been used to prove the stability of traveling waves of nonmonotone equations, see Chern et al. [3], Lin et al. [10], Wu et al. [22], Yang et al. [24]. In particular, Tian et al. [20] and Yang et al. [26], respectively, applied this method to prove the local stability of traveling waves for nonmonotone traveling waves for spatially discrete reaction-diffusion equations with time delay. Later, Yang and Zhang [25] established the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Unfortunately, the local stability (the initial perturbation around the traveling wave is properly small in a weighted norm) of traveling waves can only be obtained. Very recently, Mei et al. [15] developed a new method to prove the global stability of the oscillatory traveling waves of local Nicholson's blowflies equations. This method is based on some key observations for the structure of the govern equations and the anti-weighted energy method together with the Fourier transform. Later on, Zhang [28] and Xu et al. [23], respectively, applied this method successfully to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. More recently, Su and Zhang [17] further studied a discrete diffusion equation with a monostable convolution type nonlinearity and established the global stability of traveling waves with large speed. Motivated by the works [15,28,23,17,18], in this paper, we shall extend this method to study the global stability of traveling waves of spatial discrete diffusion system (1) without quasi-monotonicity.
The rest of this paper is organized as follows. In Section
In this section, we first give the equivalent integral form of the initial value problem of (1) with (2), then recall the existence of traveling waves of (1), and finally state the main result on the global stability of traveling waves of (1). Throughout this paper, we assume
First of all, we consider the initial value problem (1) with (2), i.e.,
{∂tv1(x,t)=d1D[v1](x,t)−αv1(x,t)+h(v2(x,t−τ)),∂tv2(x,t)=d2D[v2](x,t)−βv2(x,t)+g(v1(x,t−τ)),vi(x,s)=vi0(x,s), x∈R, s∈[−τ,0], i=1,2. | (6) |
According to [8], with aid of modified Bessel functions, the solution to the initial value problem
{∂tu(x,t)=d[u(x+1,t)−2u(x,t)+u(x−1,t)], x∈R, t>0,u(x,0)=u0(x), x∈R, |
can be expressed by
u(x,t)=(S(t)u0)(x)=e−2dt∞∑m=−∞Im(2dt)u0(x−m), |
where
Im(t)=∞∑k=0(t/2)m+2kk!(m+k)!, |
and
I′m(t)=12[Im+1(t)+Im−1(t)], ∀t>0,m∈Z, | (7) |
and
e−t∞∑m=−∞Im(t)=e−t[I0(t)+2I1(t)+2I2(t)+I3(t)+⋯]=1. | (8) |
Thus, the solution
{v1(x,t)=e−(2d1+α)t∑∞m=−∞Im(2d1t)v10(x−m,0)+∑∞m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))(h(v2(x−m,s−τ)))ds,v2(x,t)=e−(2d2+β)t∑∞m=−∞Im(2d2t)v20(x−m,0)+∑∞m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))(g(v1(x−m,s−τ)))ds. | (9) |
In fact, by [8,Lemma 2.1], we can differentiate the series on
∂tv1(x,t)=−(2d1+α)e−(2d1+α)t∞∑m=−∞Im(2d1t)v10(x−m,0) +e−(2d1+α)t∞∑m=−∞2d1I′m(2d1t)v10(x−m,0) +∞∑m=−∞Im(0)(h(v2(x−m,t−τ))) −(2d1+α)∞∑m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))(h(v2(x−m,s−τ)))ds +∞∑m=−∞∫t0e−(2d1+α)(t−s)2d1I′m(2d1(t−s))(h(v2(x−m,s−τ)))ds=d1[v1(x+1,t)−2v1(x,t)+v1(x−1,t)]−αv1(x,t)+h(v2(x,t−τ)) |
and
∂tv2(x,t)=−(2d2+β)e−(2d2+β)t∞∑m=−∞Im(2d2t)v20(x−m,0) +e−(2d2+β)t∞∑m=−∞2d2I′m(2d2t)v20(x−m,0) +∞∑m=−∞Im(0)(g(v1(x−m,t−τ))) −(2d2+β)∞∑m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))(g(v1(x−m,s−τ)))ds +∞∑m=−∞∫t0e−(2d2+β)(t−s)2d2I′m(2d2(t−s))(g(v1(x−m,s−τ)))ds=d2[v2(x+1,t)−2v2(x,t)+v2(x−1,t)]−βv2(x,t)+g(v1(x,t−τ)). |
Next we investigate the characteristic roots of the linearized system for the wave profile system (3) at the trivial equilibrium
P1(c,λ):=f1(c,λ)−f2(c,λ) |
for
f1(c,λ):=Δ1(c,λ)Δ2(c,λ),f2(c,λ):=h′(0)g′(0)e−2cλτ, |
with
Δ1(c,λ)=d1(eλ+e−λ−2)−cλ−α,Δ2(c,λ)=d2(eλ+e−λ−2)−cλ−β. |
It is easy to see that
Similar to [27,Lemma 3.1], we can obtain the following result.
Lemma 2.1. There exists a positive constant
Furthermore, we show the existence of traveling wave of (1). When system (1) is a quasi-monotone system, the existence of traveling wavefronts follows from [6,Theorem 1.1]. When system (1) is a non-quasi-monotone system, the existence of traveling waves can also be obtained by using auxiliary equations and Schauder's fixed point theorem [21,24], if we assume the following assumptions:
(i)
(ii)
(iii)
0<g−(u)≤g(u)≤g+(u)≤g′(0)u for u∈[0,K+1],0<h−(v)≤h(v)≤h+(v)≤h′(0)v for v∈[0,K+2]. |
Proposition 1. Assume that
K−1≤lim infξ→+∞ϕ1(ξ)≤lim supξ→+∞ϕ1(ξ)≤K+1,0≤lim infξ→+∞ϕ2(ξ)≤lim supξ→+∞ϕ2(ξ)≤K+2. |
Finally, we shall state the stability result of traveling waves derived in Proposition 1. Before that, let us introduce the following notations.
Notations.
||f||L1w(I)=∫Iw(x)|f(x)|dx, |
||f||Wk,1w(I)=k∑i=0∫Iw(x)|dif(x)dxi|dx. |
Let
F[f](η)=ˆf(η)=∫Re−ixηf(x)dx |
and the inverse Fourier transform is given by
F−1[ˆf](x)=12π∫Reixηˆf(η)dη, |
where
To guarantee the global stability of traveling waves of (1), we need the following additional assumptions.
limx→±∞(v10(x,s),v20(x,s))=(v1±,v2±) uniformly in s∈[−τ,0]. |
Consider the following function
P2(λ,c)=d2(eλ+e−λ−2)−cλ−β+max{h′(0),g′(0)}e−λcτ. |
Since
We select the weight function
w(ξ)=e−2λξ, |
where
Theorem 2.2. (Global stability of traveling waves). Assume that
vi0(x,s)−ϕi(x+cs)∈Cunif[−τ,0]∩C([−τ,0];W1,1w(R)), i=1,2,∂s(vi0−ϕi)∈L1([−τ,0];L1w(R)), i=1,2, |
then there exists
supx∈R|vi(x,t)−ϕi(x+ct)|≤Ce−μt,t>0, |
where
Cunif[r,T]={u∈C([r,T]×R)such thatlimx→+∞v(x,t)exists uniformly int∈[r,T]}. |
This section is devoted to proving the stability theorem, i.e., Theorem 2.2. Let
{Vi(ξ,t):=vi(x,t)−ϕi(x+ct)=vi(ξ−ct,t)−ϕi(ξ), i=1,2,Vi0(ξ,s):=vi0(x,s)−ϕi(x+cs)=vi0(ξ−cs,s)−ϕ(ξ), i=1,2. |
Then it follows from (1) and (3) that
{V1t+cV1ξ−d1D[V1]+αV1=Q1(V2(ξ−cτ,t−τ)),V2t+cV2ξ−d2D[V2]+βV2=Q2(V1(ξ−cτ,t−τ)),Vi(ξ,s)=Vi0(ξ,s), (ξ,s)∈R×[−τ,0], i=1,2. | (10) |
The nonlinear terms
{Q1(V2):=h(ϕ2+V2)−h(ϕ2)=h′(˜ϕ2)V2,Q2(V1):=g(ϕ1+V1)−g(ϕ1)=g′(˜ϕ1)V1, | (11) |
for some
We first prove the existence and uniqueness of solution
Lemma 3.1. Assume that
Proof. Let
{U1t−d1D[U1]+αU1=Q1(U2(x,t−τ)),U2t−d2D[U2]+βU2=Q2(U1(x,t−τ)),Ui(x,s)=vi0(x,s)−ϕi(x+cs):=Ui0(x,s), (x,s)∈R×[−τ,0], i=1,2. | (12) |
Thus, the global existence and uniqueness of solution of (10) are transformed into that of (12).
When
{U1(x,t)=e−(2d1+α)t∑∞m=−∞Im(2d1t)U10(x−m,0) +∑∞m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))Q1(U20(x−m,s−τ))ds,U2(x,t)=e−(2d2+β)t∑∞m=−∞Im(2d2t)U20(x−m,0) +∑∞m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))Q2(U10(x−m,s−τ))ds | (13) |
for
Since
limx→+∞U1(x,t)=e−(2d1+α)t∞∑m=−∞Im(2d1t)limx→+∞U10(x−m,0)+∞∑m=−∞∫t0e−(2d1+α)(t−s)Im(2d1(t−s))limx→+∞Q1(U20(x−m,s−τ))ds=e−αtU10(∞,0)+∫t0e−α(t−s)Q1(U20(∞,s−τ))∞∑m=−∞e−2d1(t−s)Im(2d1(t−s))ds=:U1(t) uniformly in t∈[0,τ] | (14) |
and
limx→+∞U2(x,t)=e−(2d2+β)t∞∑m=−∞Im(2d2t)limx→+∞U20(x−m,0) |
+∞∑m=−∞∫t0e−(2d2+β)(t−s)Im(2d2(t−s))limx→+∞Q2(U10(x−m,s−τ))ds=e−βtU20(∞,0)+∫t0e−β(t−s)Q2(U10(∞,s−τ))∞∑m=−∞e−2d2(t−s)Im(2d2(t−s))ds=:U2(t) uniformly in t∈[0,τ], | (15) |
where we have used (8). Thus, we obtain that
When
U1(x,t)=e−(2d1+α)(t−τ)∞∑m=−∞Im(2d1(t−τ))U1(x−m,τ)+∞∑m=−∞∫tτe−(2d1+α)(t−s)Im(2d1(t−s))Q1(U2(x−m,s−τ))ds,U2(x,t)=e−(2d2+β)(t−τ)∞∑m=−∞Im(2d2(t−τ))U2(x−m,τ)+∞∑m=−∞∫tτe−(2d2+β)(t−s)Im(2d2(t−s))Q2(U1(x−m,s−τ))ds. |
Similarly, by (14) and (15), we have
limx→+∞U1(x,t)=e−(2d1+α)(t−τ)∞∑m=−∞Im(2d1(t−τ))limx→+∞U1(x−m,τ)+∞∑m=−∞∫tτe−(2d1+α)(t−s)Im(2d1(t−s))limx→+∞Q1(U2(x−m,s−τ))ds=e−α(t−τ)U1(τ)+∫tτe−α(t−s)Q1(U1(s−τ))∞∑m=−∞e−2d1(t−s)Im(2d1(t−s))ds=:ˉU1(t) uniformly in t∈[τ,2τ], |
and
limx→+∞U2(x,t)=e−(2d2+β)(t−τ)∞∑m=−∞Im(2d2(t−τ))limx→+∞U2(x−m,τ)+∞∑m=−∞∫tτe−(2d2+β)(t−s)Im(2d2(t−s))limx→+∞Q2(U1(x−m,s−τ))ds=e−β(t−τ)U2(τ)+∫tτe−β(t−s)Q2(U2(s−τ))∞∑m=−∞e−2d2(t−s)Im(2d2(t−s))ds |
=:ˉU2(t) uniformly in t∈[τ,2τ]. |
By repeating this procedure for
Now we state the stability result for the perturbed system (10), which automatically implies Theorem 2.2.
Proposition 2. Assume that
Vi0∈Cunif[−τ,0]∩C([−τ,0];W1,1w(R)), i=1,2, |
and
∂sVi0∈L1([−τ,0];L1w(R)), i=1,2, |
then there exists
supξ∈R|Vi(ξ,t)|≤Ce−μt,t>0, i=1,2, | (16) |
for some
In order to prove Proposition 2, we first investigate the decay estimate of
Lemma 3.2. Assume that
supξ∈[x0,+∞)|Vi(ξ,t)|≤Ce−μ1t, t>0, i=1,2, |
for some
Proof. Denote
z+i(t):=Vi(∞,t), z+i0(s):=Vi0(∞,s), s∈[−τ,0], i=1,2. |
Since
limξ→+∞Vi(ξ,t)=z+i(t) |
exists uniformly for
{dz+1dt+αz+1−h′(v2+)z+2(t−τ)=P1(z+2(t−τ)),dz+2dt+βz+2−g′(v1+)z+1(t−τ)=P2(z+1(t−τ)),z+i(s)=z+i0(s), s∈[−τ,0], i=1,2, |
where
{P1(z+2)=h(v2++z+2)−h(v2+)−h′(v2+)z+2,P2(z+1)=g(v1++z+1)−g(v1+)−g′(v1+)z+1. |
Then by [9,Lemma 3.8], there exist positive constants
|Vi(∞,t)|=|z+i(t)|≤Ce−μ1t, t>0, i=1,2, | (17) |
provided that
By the continuity and the uniform convergence of
supξ∈[x0,+∞)|Vi(ξ,t)|≤Ce−μ1t, t>0, i=1,2, |
provided that
limx→+∞(v10(x,s),v20(x,s))=(K1,K2) uniformly in s∈[−τ,0], |
which implies
limξ→+∞Vi0(ξ,s)=limξ→+∞[vi0(ξ,s)−ϕi(ξ)]=Ki−Ki=0 |
uniformly for
Next we are going to establish the a priori decay estimate of
˜Vi(ξ,t)=√w(ξ)Vi(ξ+x0,t)=e−λξVi(ξ+x0,t),i=1,2. |
Substituting
{˜V1t+c˜V1ξ+c1˜V1(ξ,t)−d1eλ˜V1(ξ+1,t)−d1e−λ˜V1(ξ−1,t)=˜Q1(˜V2(ξ−cτ,t−τ)),˜V2t+c˜V2ξ+c2˜V2(ξ,t)−d2eλ˜V2(ξ+1,t)−d2e−λ˜V2(ξ−1,t)=˜Q2(˜V1(ξ−cτ,t−τ)),˜Vi(ξ,s)=√w(ξ)Vi0(ξ+x0,s)=:˜Vi0(ξ,s), ξ∈R,s∈[−τ,0], i=1,2, | (18) |
where
c1=cλ+2d1+α,c2=cλ+2d2+β |
and
˜Q1(˜V2)=e−λξQ1(V2),˜Q2(˜V1)=e−λξQ2(V1). |
By (11),
˜Q1(˜V2(ξ−cτ,t−τ))=e−λξQ1(V2(ξ−cτ+x0,t−τ))=e−λξh′(˜ϕ2)V2(ξ−cτ+x0,t−τ)=e−λcτh′(˜ϕ2)˜V2(ξ−cτ,t−τ) | (19) |
and
˜Q2(˜V1(ξ−cτ,t−τ))=e−λcτg′(˜ϕ1)˜V1(ξ−cτ,t−τ). | (20) |
By (H3), we further obtain
|˜Q1(˜V2(ξ−cτ,t−τ))|≤h′(0)e−λcτ|˜V2(ξ−cτ,t−τ)|,|˜Q2(˜V1(ξ−cτ,t−τ))|≤g′(0)e−λcτ|˜V1(ξ−cτ,t−τ)|. |
Taking (19) and (20) into (18), one can see that the coefficients
{V+1t+cV+1ξ+c1V+1(ξ,t)−d1eλV+1(ξ+1,t)−d1e−λV+1(ξ−1,t) =h′(0)e−λcτV+2(ξ−cτ,t−τ),V+2t+cV+2ξ+c2V+2(ξ,t)−d2eλV+2(ξ+1,t)−d2e−λV+2(ξ−1,t) =g′(0)e−λcτV+1(ξ−cτ,t−τ), | (21) |
with
V+i(ξ,s)=√w(ξ)Vi0(ξ+x0,s)=:V+i0(ξ,s), i=1,2, |
where
Now we are in a position to derive the decay estimate of
Lemma 3.3. ([11,Lemma 3.1]) Let
{ddtz(t)=Az(t)+Bz(t−τ),t≥0,τ>0,z(s)=z0(s),s∈[−τ,0]. | (22) |
where
z(t)=eA(t+τ)eB1tτz0(−τ)+∫0−τeA(t−s)eB1(t−τ−s)τ[z′0(s)−Az0(s)]ds, |
where
eB1tτ={0,−∞<t<−τ,I,−τ≤t<0,I+B1t1!,0≤t<τ,I+B1t1!+B21(t−τ)22!,τ≤t<2τ,⋮⋮I+B1t1!+B21(t−τ)22!+⋯+Bm1[t−(m−1)τ]mm!,(m−1)τ≤t<mτ,⋮⋮ |
where
Lemma 3.4. ([11,Theorem 3.1]) Suppose
‖z(t)‖≤C0e−ετσt,t>0, |
where
‖eAteB1tτ‖≤C0e−ετσt,t>0, |
where
From the proof of [11,Theome 3.1], one can see that
μ1(A)=limθ→0+‖I+θA‖−1θ=max1≤j≤N[Re(ajj)+N∑j≠i|aij|] |
and
μ∞(A)=limθ→0+‖I+θA‖∞−1θ=max1≤i≤N[Re(aii)+N∑i≠j|aij|]. |
Taking the Fourier transform to (21) and denoting the Fourier transform of
{∂∂tˆV+1(η,t)=(−c1+d1(eλ+iη+e−(λ+iη))−icη)ˆV+1(η,t) +h′(0)e−cτ(λ+iη)ˆV+2(η,t−τ),∂∂tˆV+2(η,t)=(−c2+d2(eλ+iη+e−(λ+iη))−icη)ˆV+2(η,t) +g′(0)e−cτ(λ+iη)ˆV+1(η,t−τ),ˆV+i(η,s)=ˆV+i0(η,s), η∈R, s∈[−τ,0], i=1,2. | (23) |
Let
A(η)=(−c1+d1(eλ+iη+e−(λ+iη))−icη00−c2+d2(eλ+iη+e−(λ+iη))−icη) |
and
B(η)=(0h′(0)e−cτ(λ+iη)g′(0)e−cτ(λ+iη)0). |
Then system (23) can be rewritten as
ˆV+t(η,t)=A(η)ˆV+(η,t)+B(η)ˆV+(η,t−τ). | (24) |
By Lemma 3.3, the linear delayed system (24) can be solved by
ˆV+(η,t)=eA(η)(t+τ)eB1(η)tτˆV+0(η,−τ)+∫0−τeA(η)(t−s)eB1(η)(t−s−τ)τ[∂sˆV+0(η,s)−A(η)ˆV+0(η,s)]ds:=I1(η,t)+∫0−τI2(η,t−s)ds, | (25) |
where
(26) |
(27) |
Lemma 3.5. Let the initial data
Then
where
Proof. According to (26), we shall estimate
where
since
By considering
Furthermore, we obtain
where
(28) |
where
Applying (28), we derive
(29) |
with
Note that
Similarly, we can obtain
It then follows that
(30) |
Substituting (29) and (30) to (26), we obtain the following the decay rate
This proof is complete.
The following maximum principle is needed to obtain the crucial boundedness estimate of
Lemma 3.6. Let
(31) |
then
Lemma 3.7. When
Proof. When
(32) |
Applying (32) to the first equation of (21), we get
By Lemma 3.6, we derive
(33) |
Similarly, we obtain
Using Lemma 3.6 again, we obtain
(34) |
When
(35) |
Combining (33), (34) and (31), we obtain
Now we establish the following crucial boundedness estimate for
Lemma 3.8. Let
(36) |
then
Proof. First of all, we prove
(37) |
Then by
(38) |
and
(39) |
Let
We are going to estimate
From (18), (19), (21) and (38), we see that
By Lemma 3.6, we obtain
namely,
(40) |
Similarly, one has
Applying Lemma 3.6 again, we have
i.e.,
(41) |
On the other hand,
Then Lemma 3.6 implies that
that is,
(42) |
Similarly,
Therefore, we can prove that
namely
(43) |
Combining (40) and (42), we obtain
(44) |
and combining (41) and (43), we prove
(45) |
Next, when
Repeating this procedure, we then further prove
which implies
The proof is complete.
Let us choose
and
Combining Lemmas 3.5 and 3.8, we can get the convergence rates for
Lemma 3.9. When
for some
Lemma 3.10. It holds that
for some
Proof. Since
which implies
Thus, the estimate for the unshifted
Proof of Proposition 3.2. By Lemmas 3.2 and 3.10, we immediately obtain (16) for
We are grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.
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