Citation: Jorge E. Macías-Díaz, Anastasios Bountis, Helen Christodoulidi. Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials[J]. Mathematics in Engineering, 2019, 1(2): 343-358. doi: 10.3934/mine.2019.2.343
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In this paper, we mainly focus on the following stream-population model[1]
{∂u∂t=d∂2u∂x2−α∂u∂x−σu+μv,∂v∂t=ϵ∂2v∂x2+σu−μv+f(v), | (1.1) |
with the initial data
u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,∀x∈R. | (1.2) |
Here, u(x,t), v(x,t) are the population densities of neuston and benthon respectively; σ represents the per capita drift rate at which the species returns to the benthic population; μ is the per capita rate at which individuals in the benthic population enter the drift; α is the speed of the flow; d, ϵ are the diffusion coefficients of species u, v respectively; the reaction term f(v) is a strictly convex function of second order differentiable satisfying f(0)=f(1)=0, f′(0)>0>f′(1), and f(v)>0 for v∈(0,1). σ,μ,α,d,ϵ are positive constants and possess the biological meaning[2,3]. It is worth mentioning that the benthon basically do not move horizontally since ϵ≪1. Obviously, (0,0) and (μσ,1) are the equilibria and further we can obtain that (0,0) is unstable and (μσ,1) is stable for the corresponding spatially homogeneous system of (1.1).
In this paper, we are interested in the nonnegative traveling wave, connecting (0,0) and (μσ,1), which possesses the wave profile as
u(x,t)=¯U(z),v(x,t)=¯V(z),z=x−ct, | (1.3) |
where the wave speed c≥0. Substituting (1.3) into (1.1), yields the new wave profile as
{−c¯U′=d¯U″−α¯U′−σ¯U+μ¯V,−c¯V′=ϵ¯V″+σ¯U−μ¯V+f(¯V), | (1.4) |
with the boundary value condition
(¯U,¯V)(−∞)=(μσ,1),(¯U,¯V)(∞)=(0,0). | (1.5) |
It is worth pointing out that the existence of the traveling wave solution of (1.4) was proved, and the determinacy of linear and nonlinear selection of the minimal wave speed was further established by employing the method of upper and lower solutions in [1].
On this basis of the existence, we shall investigate the local and the global stability of the monotone traveling waves. In this progress, we need to conform that the solution of (1.1) is infinitely close to (¯U,¯V)(x−ct) when t is large enough for given initial data u0(x) and v0(x). To begin with, we set
(u,v)(x,t)=(U,V)(z,t), | (1.6) |
and then (1.4) can be transformed into the partial differential model
{Ut=dUzz+(c−α)Uz−σU+μV,Vt=ϵVzz+cVz+σU−μV+f(V), | (1.7) |
subject to
U(z,0)=u0(z),V(z,0)=v0(z),∀z∈R. | (1.8) |
From the above system (1.7) we know that (¯U,¯V)(z) is also its steady state.
The investigation of the stabilities of traveling wave solutions could be traced back to the original works [4,5]. Then many works assessing stability are available, but differences among them are frequent. For examples, the stability of the traveling waves with critical speed [6] and noncritical speed [7] in appropriate weighted Banach spaces is proved. The global stability of the forced waves [8,9] is presented. The time-periodic traveling waves are asymptotically stable [10,11]. Mei et. al. [12,13,14] proved the exponential stability of traveling wave fronts, and the exponential stability of stochastic systems is demonstrated [15,16]. By analyzing the location of the spectrum, the local stability of the traveling waves are investigated [17,18]. For more related works, we refer to [19,20,21,22,23,24,25,26,27,28,29,30,31,32], and the references cited therein.
Despite the success in the study of the existence and the speed determinacy of traveling waves to the model (1.1), the local and global stability of the traveling waves in the monostable still remains unsolved. In this paper, mainly inspired by the ideas in [18], we devote our attention to finishing the issue of stability for the steady state (¯U,¯V)(z). Firstly, the local stability is discussed on the basis of asymptotic behavior of the traveling waves near the equilibrium and the method of spectrum analysis. When analyzing the eigenvalue problem, we need to overcome the great challenge brought by the high-order nonlinear terms. Then by using the result of local stability, we should choose appropriate weighted Banach space. Furthermore, also by constructing upper and lower solutions, we prove that the solution (U,V) in the moving coordinates converges to the steady state. The new results on the global stability is established in the special weighted function space by using the squeeze theorem.
This paper are organized as follows. In section 2, we present some preliminaries and main results. In section 3, we prove the local stability of the traveling waves. The global stability of the steady state (¯U,¯V)(z) in a special choice on the weighted function space L∞ω(R) is investigated in section 4.
First, linearizing the system (1.4) at (0,0) we obtain
{−c¯U′=d¯U″−α¯U′−σ¯U+μ¯V,−c¯V′=ϵ¯V″+σ¯U−μ¯V+f′(0)¯V. | (2.1) |
The transformation
(¯U,¯V)(z)∼(ξ1e−λz,ξ2e−λz) | (2.2) |
with λ>0 and ξi(i=1,2) being positive constants, allows us to transform the discussion of the asymptotic behaviors at infinitely in variable z to the following eigenvalue problem
cλξ=(dλ2+αλ−σμσϵλ2−μ+f′(0))ξ, | (2.3) |
where ξ=(ξ1,ξ2)T. Letting
A1(λ)=dλ2+αλ−σ,A2(λ)=ϵλ2−μ+f′(0),A(λ)=(A1(λ)μσA2(λ)), | (2.4) |
then the eigenvalue problem (2.3) is equivalent to r(λ)ξ=A(λ)ξ. Further, we can obtain the eigenvalues
r±(λ)=A1(λ)+A2(λ)±√(A1(λ)−A2(λ))2+4σμ2. | (2.5) |
As we know, the principal eigenvalue is greater than or equal to the real part of matrix eigenvalues. According to [33], the principal eigenvalue of matrix A(λ) is
r(λ)=r+(λ), | (2.6) |
where r+ is real number and for any λ∈(0,+∞),
r+(λ)=12[A1(λ)+A2(λ)+√(A1(λ)+A2(λ))2−4(A1(λ)A2(λ)−σμ)]=12[(d+ϵ)λ2+αλ−σ−μ+f′(0)+√[(d−ϵ)λ2+αλ−σ+μ−f′(0)]2+4σμ]>0. | (2.7) |
Actually, through the classified discussion of the positive and negative of A1and A2, r+(λ)>0 holds true.
Lemma 2.1. ([1], see Section 2) r(λ), defined in (2.7), is a real, continuous, and convex function with respect to λ∈R. Assume that
c0=infλ∈(0,∞)r(λ)λ∈R+, | (2.8) |
where c0 is called the minimal wave speed, then we can obtain that the equation cλ=r(λ) has
● no solution, if c<c0;
● a unique solution λ(c0), if c=c0;
● two solutions λ1(c) and λ2(c) with λ1(c)<λ2(c), if c>c0.
Based on the above analysis, we could give the asymptotic behaviors as z→∞ with the following lemma.
Lemma 2.2. ([1], see Section 2) The asymptotic behaviors of the traveling wave solution (¯U(z),¯V(z)) (see (2.1)) can be represented as follows:
(¯U¯V)=C1(ξ1(λ1(c))ξ2(λ1(c)))e−λ1(c)z+C2(ξ1(λ2(c))ξ2(λ2(c)))e−λ2(c)z | (2.9) |
with C1>0 or C1=0,C2>0, where the eigenvectors corresponding to eigenvalues λi(c)(i=1,2) are
(ξ1(λi(c))ξ2(λi(c)))=(μcλi(c)−A1(λi(c)))or(cλi(c)−A2(λi(c))σ). | (2.10) |
The proof of the above lemmas can refer to the proof of Theorem 1 in [18], which will not be repeated here.
Before stating our main results, let us make the following notation.
Notation: Throughout the paper, Lp are function spaces defined by using a natural generalization of the p-norm for finite dimensional vector spaces. The weighted Lebesgue space Lpω with 1≤p<∞ can be expressed by
Lpω={g(z):g(z)ω(z)∈Lp(R)} | (2.11) |
with the norm
‖g(z)‖Lpω=(∫∞−∞|g(z)|p|ω(z)|pdz)1p, | (2.12) |
where the weighted function
ω(z)={e−β(z−z0),z>z0,1,z≤z0 | (2.13) |
for some positive constants z0 and β.
With the introduction above, we state our main conclusions for two theorems.
Theorem 1. (Local stability) For any c>c0, the wavefront (¯U,¯V)(z) is locally stable in the weighted function space Lpω if β∈(λ1,λ2) to be chosen.
Theorem 2. (Global stability) Assume that c>c0, λ1<β<λ2 and the initial data U(z,0)=U0(z) and V(z,0)=V0(z) satisfy
(0,0)≤(U0,V0)(z)≤(μσ,1), ∀z∈R,lim inf |
and
|U_0(z)-\overline{U}(z)|\in L_\omega^\infty(\mathbb{R}), |V_0(z)-\overline{V}(z)|\in L_\omega^\infty(\mathbb{R}). |
Then (1.7) with initial data admits a unique solution (U, V)(z, t) satisfying
(0, 0)\leq(U, V)(z, t)\leq\left(\frac{\mu}{\sigma}, 1\right), \forall (z, t)\in \mathbb{R}\times\mathbb{R}^+. |
Moreover, the inequalities
\mathop{\sup}\limits_{z\in\mathbb{R}}|U(z, t)-\overline{U}(z)|\leq k e^{-\eta t}, t > 0 |
and
\mathop{\sup}\limits_{z\in\mathbb{R}}|V(z, t)-\overline{V}(z)|\leq k e^{-\eta t}, t > 0 |
hold for some positive constants k and \eta .
The purpose in this section is to apply the weighted energy method and spectral analysis to prove the local stability of the traveling waves presented in Theorem 1.
First, we assume that
\begin{equation} U(z, t) = \overline{U}(z)+\delta\psi_{1}(z)e^{\chi t}, V(z, t) = \overline{V}(z)+\delta\psi_{2}(z)e^{\chi t}, \end{equation} | (3.1) |
where \delta\ll1 , \chi is a parameter, and \psi_{1}(z) and \psi_{2}(z) are real functions. Substituting (U(z, t), V(z, t)) into system (1.7), and linearizing the new system with respect to (\overline{U}, \overline{V})(z) , we obtain the following spectral problem:
\begin{equation} \chi\Psi = \mathfrak{L}\Psi: = D\Psi''+C\Psi'+B\Psi, \end{equation} | (3.2) |
where
\begin{equation} \Psi = (\psi_{1}, \psi_{2})^{T}, D = \left( \begin{array}{cc} d & 0 \\ 0 & \epsilon \\ \end{array} \right), C = \left( \begin{array}{cc} c-\alpha & 0 \\ 0 & c \\ \end{array} \right), B = \left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu+f'(\overline{V}) \\ \end{array} \right), \end{equation} | (3.3) |
where the sign of the maximal real part to the spectrum of the operator \mathfrak{L} is related to the local stability of the traveling wave solution. To proceed, we set
\begin{equation} \Psi = \left( \begin{array}{c} \psi_{1} \\ \psi_{2} \\ \end{array} \right) = \left( \begin{array}{c} \omega\phi_1 \\ \omega\phi_2 \\ \end{array} \right), \end{equation} | (3.4) |
where \phi_1 and \phi_2 are the functions in the space L^p , \omega has been defined in (2.13) with \lambda_1 < \beta < \lambda_2 . Substitute (3.4) into (3.2) to obtain a new spectral problem as:
\begin{equation} \chi\Phi = \mathfrak{L}_\omega\Phi: = D\Phi''+H\Phi'+J\Phi, \end{equation} | (3.5) |
where \Phi = (\phi_{1}, \phi_{2})^{T} , D is defined in (3.3) and
\begin{equation} \begin{aligned} & H = \left( \begin{array}{cc} c-\alpha+2d\frac{\omega'}{\omega} & 0 \\ 0 & c+2\epsilon\frac{\omega'}{\omega} \\ \end{array} \right), \\ & J = \left( \begin{array}{cc} d\frac{\omega''}{\omega}+(c-\alpha)\frac{\omega'}{\omega}-\sigma & \mu \\ \sigma & \epsilon\frac{\omega''}{\omega}+c\frac{\omega'} {\omega}-\mu+f'(\overline{V}) \\ \end{array} \right). \end{aligned} \end{equation} | (3.6) |
Since H(z) and J(z) are bounded real matrix functions and \omega , \omega' , \omega'' exist as z\rightarrow \pm\infty , we define
\begin{equation} H_\pm = \mathop{\lim}\limits_{z\rightarrow \pm\infty}H(z), J_\pm = \mathop{\lim}\limits_{z\rightarrow \pm\infty}J(z). \end{equation} | (3.7) |
Further, we analyze the position of the essential spectrum for the operator \mathfrak{L}_\omega and give the following lemma.
Lemma 3.1. ([34], see Theorem A.2) If we define
\begin{equation} S_\pm = \{\chi|det(-\theta^2 D+i\theta H_\pm+J_\pm-\chi I) = 0, \; -\infty < \theta < \infty\}, \end{equation} | (3.8) |
then the essential spectrum of \mathfrak{L}_\omega is contained in the union of the regions inside or on the curves S_+ and S_- , which are on the left-half complex plane, when the condition \lambda_1 < \beta < \lambda_2 is satisfied.
Proof. We prove the lemma 3.1 for two cases with respect to z .
Case 1. When z\rightarrow +\infty in (3.6), we have
\begin{equation} \begin{aligned} & H_+ = \mathop{\lim}\limits_{z\rightarrow +\infty}H(z) = \left( \begin{array}{cc} c-\alpha-2d\beta & 0 \\ 0 & c-2\epsilon\beta \\ \end{array} \right), \\ & J_+ = \mathop{\lim}\limits_{z\rightarrow +\infty}J(z) = \left( \begin{array}{cc} d\beta^2-(c-\alpha)\beta-\sigma & \mu \\ \sigma & \epsilon\beta^2-c\beta-\mu+f'(0) \\ \end{array} \right). \end{aligned} \end{equation} | (3.9) |
Therefore, the equation det(-\theta^2 D+i\theta H_++J_+-\chi I) = 0 has two solutions, i.e.
\begin{equation} \chi_{1, 2} = \frac{1}{2}\left\{(E+G)+i(F+K)\pm\sqrt{[(E-G)+i(F-K)]^2+4\sigma\mu}\right\} \end{equation} | (3.10) |
with E = -d\theta^2+A_1(\beta)-c\beta , F = (c-\alpha-2d\beta)\theta , G = -\epsilon\theta^2+A_2(\beta)-c\beta , K = (c-2\epsilon\beta)\theta , where A_1 and A_2 have been noted in (2.4). Assume that
\begin{equation} S_{+, 1} = \{\chi_1|-\infty < \theta < \infty\}, S_{+, 2} = \{\chi_2|-\infty < \theta < \infty\}. \end{equation} | (3.11) |
By Euler's formula, we obtain the real parts of the eigenvalues \chi_{1, 2} as follows:
\begin{equation} \begin{aligned} Re(\chi_{1}) & = \frac{1}{2}\bigg[(E+G)\\ &+\sqrt{\frac{1}{2}\left(\sqrt{((E-G)^2+(F-K)^2+4\sigma\mu)^2-16(F-K)^2\sigma\mu} +(E-G)^2-(F-K)^2+4\sigma\mu\right)}\bigg], \\ Re(\chi_{2})& = \frac{1}{2}\bigg[(E+G)\\ &-\sqrt{\frac{1}{2}\left(\sqrt{((E-G)^2+(F-K)^2+4\sigma\mu)^2-16(F-K)^2\sigma\mu} +(E-G)^2-(F-K)^2+4\sigma\mu\right)}\bigg]. \end{aligned} \end{equation} | (3.12) |
Obviously, Re(\chi_{1}) > Re(\chi_{2}) . By a simple calculation, one can obtain that
\begin{equation} \begin{aligned} Re(\chi_{1}) & < \frac{1}{2}\left[(E+G)+\sqrt{(E-G)^2+4\sigma\mu}\right] \\ & = \frac{1}{2}\left[-(d+\epsilon)\theta^2+A_1(\beta) +A_2(\beta)+\sqrt{-(d-\epsilon)\theta^2+(A_1(\beta) -A_2(\beta))^2+4\sigma\mu}\right]-c\beta\\ &\leq\frac{1}{2}\left[A_1(\beta)+A_2(\beta) +\sqrt{(A_1(\beta)-A_2(\beta))^2+4\sigma\mu}\right]-c\beta\\ & = r_{+}(\beta)-c\beta\\ & < 0. \end{aligned} \end{equation} | (3.13) |
since the formulas (2.7 and 2.8) and \lambda_1 < \beta < \lambda_2 are satisfied for c > c_0 . That is to say S_+ = S_{+, 1}\cup S_{+, 2} is on the left-half complex plane.
Case 2. When z\rightarrow -\infty , it follows that
\begin{equation} \begin{aligned} & H_- = \mathop{\lim}\limits_{z\rightarrow -\infty}H(z) = \left( \begin{array}{cc} c-\alpha & 0 \\ 0 & c \\ \end{array} \right), \\ & J_- = \mathop{\lim}\limits_{z\rightarrow -\infty}J(z) = \left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu+f'(0) \\ \end{array} \right). \end{aligned} \end{equation} | (3.14) |
Similar to Case 1, solving the equation det(-\theta^2 D+i\theta H_-+J_–\chi I) = 0 , we obtain
\begin{equation} \chi_{3, 4} = \frac{1}{2}\left[(M+P)+i(N+Q)\pm\sqrt{((M-P)+i(N-Q))^2+4\sigma\mu}\right], \end{equation} | (3.15) |
where M = -d\theta^2-\sigma , N = (c-\alpha)\theta , P = -\epsilon\theta^2-\mu+f'(1) , Q = c\theta . Further, Euler's formulas allow us to obtain the maximal real part of \chi_{3, 4} as
\begin{equation} \begin{aligned} Re(\chi_{3})& = \frac{1}{2}\bigg[(M+P)\\ &+\sqrt{\frac{1}{2}\left(\sqrt{\left((M-P)^2+(N-Q)^2+4\sigma\mu\right)^2-16(N-Q)^2\sigma\mu} +(M-P)^2-(N-Q)^2+4\sigma\mu\right)}\bigg]\\ &\leq\frac{1}{2}\left[(M+P)+\sqrt{(M+P)^2-4(MP-\sigma\mu)}\right]\\ & < 0, \end{aligned} \end{equation} | (3.16) |
because of M+P < 0 and MP-\sigma\mu > 0 . It means that S_- = \{\chi_3|-\infty < \theta < \infty\} \cup \{\chi_4|-\infty < \theta < \infty\} is also on the left-half complex plane.
By the above analysis, the essential spectrum of the operator \mathfrak{L}_\omega is on the left-half complex plane.
If the sign of the real part of the principal eigenvalue in the point spectrum (3.2) is negative, the traveling wave solutions is locally stable. So, we need the following step to check the sign of the principal eigenvalue to ensure the result of local stability.
Finally, we judge the sign of the principal eigenvalue in the point spectrum to prove the local stability. We first discuss the asymptotic behavior of the system (1.4) as z\rightarrow -\infty . By linearizing the system (1.4) at (\frac{\mu}{\sigma}, 1) , we have
\begin{equation} \left\{ \begin{aligned} &-c\overline{U}' = d \overline{U}''-\alpha \overline{U}'-\sigma \overline{U}+\mu \overline{V}, \\ &-c\overline{V}' = \epsilon \overline{V}''+\sigma \overline{U}-\mu \overline{V}+f'(1)(\overline{V}-1). \end{aligned} \right. \end{equation} | (3.17) |
Let
\begin{equation} (\overline{U}, \overline{V})(z) = \left(\frac{\mu}{\sigma}-\xi_{3}e^{\lambda z}, 1-\xi_{4}e^{\lambda z}\right) \end{equation} | (3.18) |
with \lambda > 0 and \xi_{3} , \xi_{4} being positive constants. Substituting (3.18) into (3.17), it follows that
\begin{equation} \left( \begin{array}{cc} d\lambda^{2}+(c-\alpha)\lambda-\sigma & \mu \\ \sigma & \epsilon \lambda^{2}+c \lambda-\mu+f'(1) \\ \end{array} \right)\xi = 0, \end{equation} | (3.19) |
where \xi = (\xi_{3}, \xi_{4})^{T} . Suppose that \lambda_i (i = 3, 4, 5, 6) are the eigenvalues to the left matrix of Eq (3.19). According to the Vieta's theorem, we know these four eigenvalues are two positive numbers and two negative numbers. Without loss of generality, we assume that \lambda_3 > \lambda_4 > 0 > \lambda_5 > \lambda_6 for c > 0 . Then the wave profile has the following asymptotic behaviors:
\begin{equation} \left( \begin{array}{c} \overline{U} \\ \overline{V} \\ \end{array} \right) = \left( \begin{array}{c} \frac{\mu}{\sigma} \\ 1 \\ \end{array} \right) -C_3\left( \begin{array}{c} \xi_3(\lambda_4) \\ \xi_4(\lambda_4) \\ \end{array} \right)e^{\lambda_4z}-C_4\left( \begin{array}{c} \xi_3(\lambda_3) \\ \xi_4(\lambda_3) \\ \end{array} \right)e^{\lambda_3z}, \; as \; z \rightarrow - \infty \end{equation} | (3.20) |
with C_3 > 0 or C_3 = 0, C_4 > 0 .
To associate with (3.2), we consider the following system
\begin{equation} u_t = Du_{zz}+Cu_z+Bu, \end{equation} | (3.21) |
where u(z, t) = (u_1(z, t), u_2(z, t))^T and D , C , B are defined in (3.3). For a given solution semiflow Q_{t} = u(z, t, \psi) of (3.21) with any given initial data \phi\in L^{p} , we denote by e^{\chi t}\Psi the solution of (3.21). It is easy to see that Q_{t} is compact and strongly positive. By the Krein-Rutman theorem (see, e.g., [35]), Q_{t} has a simple principal eigenvalue \chi_{max} with a strongly positive eigenvector, and all other eigenvalues must satisfy e^{\chi t}\Psi < e^{\chi_{max} t}\Psi .
Next, we shall discuss the eigenvalue \chi for two cases as \chi = 0 and \chi > 0 .
Case 1. When \chi = 0 , it can be directly calculated that (-\overline{U}', -\overline{V}')(z) is the corresponding positive eigenvector derived form (3.2), where the asymptotic behavior of the solution of (2.1), i.e. (\overline{U}, \overline{V})(z)\sim (C_{1}e^{-\lambda_{1} z}, C_{1}e^{-\lambda_{1} z}) , C_{1} > 0 , as z\rightarrow \infty . Although it could be check that the positive eigenvector (-\overline{U}', -\overline{V}')(z) is not belong to the weighted space L_\omega^p since \lambda_{1} < \beta < \lambda_{2} .
Case 2. In this case, we discuss the eigenvalue \chi > 0 to produce a contrary for two cases with respect to z , namely, z\rightarrow -\infty and z\rightarrow \infty . Assume that \chi > 0 and \Psi\in L_\omega^p . Then we know that the minimal positive eigenvalue of the \Psi is larger than \beta . Since \overline{\Psi}(z) = (-\overline{U}', -\overline{V}')(z) is a positive solution of (3.21), it follows that \overline{\Psi}(z) > \Psi as z\rightarrow \infty .
When z\rightarrow -\infty , assume that \Psi(z) has the asymptotic behavior as k e^{\lambda z} for some positive k and \lambda . Substituting it into the spectral problem (3.5), we obtain the characteristic equation in eigenvalue \chi as follows:
\begin{equation} \left| \begin{array}{cc} d\lambda^{2}+(c-\alpha)\lambda-\sigma-\chi & \mu \\ \sigma & \epsilon \lambda^{2}+c \lambda-\mu+f'(1)-\chi \\ \end{array} \right| = 0. \end{equation} | (3.22) |
Next, we shall show the relationship between the four roots of (3.22) denoted by \hat{\lambda}_{i}(i = 3, 4, 5, 6) and \lambda_{i}(i = 3, 4, 5, 6) , namely, \hat{\lambda}_{3} > \lambda_{3} , \hat{\lambda}_{4} > \lambda_{4} , \hat{\lambda}_{5} < \lambda_{5} , \hat{\lambda}_{6} < \lambda_{6} . In fact, when \chi > 0 , the parabolic
\begin{equation} \begin{aligned} &p_{1}: d\lambda^{2}+(c-\alpha)\lambda-\sigma-\chi = 0, \\ &p_{2}: d\lambda^{2}+(c-\alpha)\lambda-\sigma = 0, \end{aligned} \end{equation} | (3.23) |
have two roots r_{p_{1}}^{\pm} and r_{p_{2}}^{\pm} with one positive and one negative, respectively. It is easy to see that r_{p_{1}}^{-} < r_{p_{2}}^{-} and r_{p_{1}}^{+} < r_{p_{2}}^{+} . Similar discussion to
\begin{equation} \begin{aligned} &p_{3}: \epsilon \lambda^{2}+c \lambda-\mu+f'(1)-\chi = 0, \\ &p_{4}: \epsilon \lambda^{2}+c \lambda-\mu+f'(1) = 0, \end{aligned} \end{equation} | (3.24) |
we also obtain that r_{p_{3}}^{-} < r_{p_{4}}^{-} and r_{p_{3}}^{+} < r_{p_{4}}^{+} . Further, we know that r_{p_{1}}^{-} , r_{p_{1}}^{+} , r_{p_{3}}^{-} and r_{p_{3}}^{+} are also the roots of
\begin{equation} p_{5}: (d\lambda^{2}+(c-\alpha)\lambda-\sigma-\chi)(\epsilon \lambda^{2} +c \lambda-\mu+f'(1)-\chi) = 0 \end{equation} | (3.25) |
and r_{p_{2}}^{-} , r_{p_{2}}^{+} , r_{p_{4}}^{-} and r_{p_{4}}^{+} are the roots of
\begin{equation} p_{6}: (d\lambda^{2}+(c-\alpha)\lambda-\sigma)(\epsilon \lambda^{2}+c \lambda-\mu+f'(1)) = 0. \end{equation} | (3.26) |
That is to say that the positive roots of p_{5} related to p_{6} are moving right. Therefore, when the curves of p_{5} and p_{6} intersects with the same line y = \mu\sigma , the points are denoted by \hat{\lambda}_{i}(i = 3, 4, 5, 6) and \lambda_{i}(i = 3, 4, 5, 6) , respectively. And the points of intersection satisfy \hat{\lambda}_{3} > \lambda_{3} , \hat{\lambda}_{4} > \lambda_{4} , \hat{\lambda}_{5} < \lambda_{5} , \hat{\lambda}_{6} < \lambda_{6} . In other words, the positive roots of \lambda are increasing with respect to \chi . This indicates that \overline{\Psi}(z)\sim k_{1}e^{\lambda_{3}z} and \Psi(z)\sim k_{2}e^{\hat{\lambda}_{3}z} as z\rightarrow -\infty .
Thus, we choose \overline{k} sufficient large such that \overline{k}\overline{\Psi}\geq|\Psi| . By the comparison principal for the system (3.21), it must be \overline{k}\overline{\Psi}(z)\geq|\Psi|e^{\chi t} . Hence the assumption \chi > 0 is incorrect. This implies that the real parts of all eigenvalues \chi of (3.5) should not be positive for \Psi\in L_\omega^p .
The proof is complete.
In this section, we will prove Theorem 2. In order to realize it, we need the following conclusion.
(Comparison principle) Let (U^+, V^+)(z, t) and (U^-, V^-)(z, t) be the solutions of (1.7) with respect to the initial values
\begin{equation} \begin{aligned} & U_0^+(z) = \max\{U_0(z), \overline{U}(z)\}, \; \; V_0^+(z) = \max\{V_0(z), \overline{V}(z)\}, \\ & U_0^-(z) = \min\{U_0(z), \overline{U}(z)\}, \; \; V_0^-(z) = \min\{V_0(z), \overline{V}(z)\} \end{aligned} \end{equation} | (4.1) |
respectively; namely
\begin{equation} \left\{\begin{aligned} &U_{t}^\pm = d U_{zz}^\pm+(c-\alpha)U_{z}^\pm-\sigma U^\pm+\mu V^\pm, \\ &V_{t}^\pm = \epsilon V_{zz}^\pm+c V_{z}^\pm+\sigma U^\pm-\mu V^\pm+f(V^\pm), \\ &(U^\pm, V^\pm)(z, 0) = (U_0^\pm, V_0^\pm)(z). \end{aligned}\right. \end{equation} | (4.2) |
It holds that
\begin{equation} \begin{aligned} & (0, 0)\leq(U_0^-, V_0^-)(z)\leq(U_0, V_0)(z)\leq(U_0^+, V_0^+)(z)\leq\left(\frac{\mu}{\sigma}, 1\right), \\ & (0, 0)\leq(U_0^-, V_0^-)(z)\leq(\overline{U}, \overline{V})(z)\leq(U_0^+, V_0^+)(z) \leq\left(\frac{\mu}{\sigma}, 1\right). \end{aligned} \end{equation} | (4.3) |
By comparison principle, we have
\begin{equation} \begin{aligned} & (0, 0)\leq(U^-, V^-)(z, t)\leq(\overline{U}, \overline{V})(z, t)\leq(U^+, V^+)(z, t)\leq \left(\frac{\mu}{\sigma}, 1\right), \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+, \\ & (0, 0)\leq(U^-, V^-)(z, t)\leq(\overline{U}, \overline{V})(z)\leq(U^+, V^+)(z, t)\leq \left(\frac{\mu}{\sigma}, 1\right), \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+. \end{aligned} \end{equation} | (4.4) |
If both (U^+, V^+)(z, t) and (U^-, V^-)(z, t) converge to (\overline{U}, \overline{V})(z) , then the squeezing theorem ensures the global stability of system (1.7) stated in (2.2).
Lemma 4.1. Under the conditions given in Theorem 2, (U^+, V^+)(z, t) converges to (\overline{U}, \overline{V})(z) .
Proof. To begin with, we suppose that
\begin{equation} R(z, t) = U^+(z, t)-\overline{U}(z), S(z, t) = V^+(z, t)-\overline{V}(z), \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+, \end{equation} | (4.5) |
which satisfies the initial values
\begin{equation} R(z, 0) = U_0^+(z)-\overline{U}(z), S(z, 0) = V_0^+(z)-\overline{V}(z). \end{equation} | (4.6) |
By (4.2) and (4.4), we have
\begin{equation} (0, 0)\leq(R, S)(z, t)\leq\left(\frac{\mu}{\sigma}, 1\right), \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+. \end{equation} | (4.7) |
Through (1.4) and (4.3), the following inequality
\begin{equation} \left( \begin{array}{c} R \\ S \\ \end{array} \right)_t\leq D\left( \begin{array}{c} R \\ S \\ \end{array} \right)_{zz}+C\left( \begin{array}{c} R \\ S \\ \end{array} \right)_z+B_{0}\left( \begin{array}{c} R \\ S \\ \end{array} \right) \end{equation} | (4.8) |
holds for the matrices C and D defined in (3.3), and
\begin{equation} B_{0} = \left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu+f'(0) \\ \end{array} \right). \end{equation} | (4.9) |
Then we consider the convergence of (R, S)(z, t) for two cases with respect to z .
Case 1. When z > z_0 , we first define
\begin{equation} \left( \begin{array}{c} R \\ S \\ \end{array} \right)(z, t) = e^{-\beta(z-z_0)}\left( \begin{array}{c} \overline{R} \\ \overline{S} \\ \end{array} \right)(z, t), \end{equation} | (4.10) |
so as to investigate the stability in weighted function space L_\omega^\infty for all (z, t)\in \mathbb{R}\times\mathbb{R}^+ , where \overline{R} and \overline{S} are the functions in L^\infty(\mathbb{R}) , the weighted function \omega(z) is defined in (3.3). Substituting (4.10) into (4.8), we obtain
\begin{equation} \begin{aligned} \left( \begin{array}{c} \overline{R} \\ \overline{S} \\ \end{array} \right)_t &\leq D\left( \begin{array}{c} \overline{R} \\ \overline{S} \\ \end{array} \right)_{zz}+\left( \begin{array}{cc} c-\alpha-2d\beta & 0 \\ 0 & c-2\epsilon\beta \\ \end{array} \right) \left( \begin{array}{c} \overline{R} \\ \overline{S} \\ \end{array} \right)_z+\left( \begin{array}{cc} A_1(\beta)-c\beta & \mu \\ \sigma & A_2(\beta)-c\beta \\ \end{array} \right) \left( \begin{array}{c} \overline{R} \\ \overline{S} \\ \end{array} \right)\\ &: = \left( \begin{array}{c} \mathfrak{L}_1(\overline{R}, \overline{S}) \\ \mathfrak{L}_2(\overline{R}, \overline{S}) \\ \end{array} \right), \end{aligned} \end{equation} | (4.11) |
where A_1(\beta) and A_2(\beta) are defined in (2.4).
Further, we choose
\begin{equation} \overline{R}_1(z, t) = k_1\zeta_1e^{-\eta_1t}, \; \overline{S}_1(z, t) = k_1\zeta_2e^{-\eta_1t}, \; \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+, \end{equation} | (4.12) |
for some positive constants k_1 and \eta_1 , where (\zeta_1, \zeta_2)^T = (\zeta_1(\beta), \zeta_2(\beta))^T is the eigenvector of matrix M_{1} . In fact,
\begin{equation} \begin{aligned} M_{1} = \left(\begin{array}{cc} A_1(\beta)-c\beta & \mu \\ \sigma & A_2(\beta)-c\beta \\ \end{array} \right) \end{aligned} \end{equation} | (4.13) |
which has the eigenvalue
\overline{\lambda}_1 = \frac{1}{2}\left(A_1(\beta)+A_2(\beta)+\sqrt{(A_1(\beta) -A_2(\beta))^2+4\sigma\mu}\right)-c\beta. |
The associated eigenvectors can be found by direct calculation as follows:
\begin{equation} \zeta_1(\beta) = \mu, \zeta_2(\beta) = \frac{1}{2}\left(-A_1(\beta)+A_2(\beta) +\sqrt{(A_1(\beta)-A_2(\beta))^2+4\sigma\mu}\right). \end{equation} | (4.14) |
It is not hard to check that, for \lambda_1 < \beta < \lambda_2 , \zeta_1(\beta) , \zeta_2(\beta) are positive and \overline{\lambda}_1 is negative. Moreover, one can obtain
\begin{equation} \mathfrak{L}_1(\overline{R}_1, \overline{S}_1) = \overline{\lambda}_1\overline{R}_1 < 0, \; \; \mathfrak{L}_2(\overline{R}_1, \overline{S}_1) = \overline{\lambda}_1\overline{S}_1 < 0. \end{equation} | (4.15) |
Thus, we can choose \eta_1\leq-\overline{\lambda}_1 such that
\begin{equation} \left( \begin{array}{c} \overline{R}_1 \\ \overline{S}_1 \\ \end{array} \right)_t = -\eta_1k_1\left( \begin{array}{c} \zeta_1 \\ \zeta_2 \\ \end{array} \right)e^{-\eta_1t}\geq\left( \begin{array}{c} \mathfrak{L}_1(\overline{R}_1, \overline{S}_1) \\ \mathfrak{L}_2(\overline{R}_1, \overline{S}_1) \\ \end{array} \right). \end{equation} | (4.16) |
Once we choose k_1\geq\mathop{\max}\limits_{z\in\mathbb{R}}\left\{\frac{\overline{R}(z, 0)}{\zeta_1}, \frac{\overline{S}(z, 0)}{\zeta_2}\right\} to make
(\overline{R}_1, \overline{S}_1)(z, 0) = (k_1\zeta_1, k_2\zeta_2)\geq(\overline{R}, \overline{S})(z, 0), |
then by comparison principle on unbounded domain, \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+ , we obtain
\begin{equation} (R, S)(z, t) = e^{-\beta(z-z_0)}(\overline{R}, \overline{S})(z, t) \leq k_1(\zeta_1, \zeta_2)e^{-\beta(z-z_0)-\eta_1t}. \end{equation} | (4.17) |
Therefore, the above inequality holds for any fixed point z_0 . That is to say, for z > z_0 , (R, S)(z, t) converges to (0, 0) .
Case 2. When z\leq z_0 , the system marked (R, S)(z, t) can be shown as
\begin{equation} \left( \begin{array}{c} R \\ S \\ \end{array} \right)_t = D\left( \begin{array}{c} R \\ S \\ \end{array} \right)_{zz}+C\left( \begin{array}{c} R \\ S \\ \end{array} \right)_z+\left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu \\ \end{array} \right) \left( \begin{array}{c} R \\ S \\ \end{array} \right)+\left( \begin{array}{c} 0 \\ f(S+\overline{V})-f(\overline{V}) \\ \end{array} \right). \end{equation} | (4.18) |
Here, f(S+\overline{V})-f(\overline{V}) is a second-order differentiable function that can be expressed as f'(1)S+h(S) ( h(S) also is second-order differentiable and h(0) = 0 ) as (\overline{U}, \overline{V})(z)\rightarrow \left(\frac{\mu}{\sigma}, 1\right) . Consequently, we need to select z_0 , so that
\begin{equation} \left( \begin{array}{c} R \\ S \\ \end{array} \right)_t \leq D\left( \begin{array}{c} R \\ S \\ \end{array} \right)_{zz}+C\left( \begin{array}{c} R \\ S \\ \end{array} \right)_z+\left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu+f'(1)+\varepsilon_1 \\ \end{array} \right) \left( \begin{array}{c} R \\ S \\ \end{array} \right)+\left( \begin{array}{c} 0 \\ h(S) \\ \end{array} \right) \end{equation} | (4.19) |
for some given enough small positive \varepsilon_1 and \varepsilon_1\ll\mu-f'(1) .
From the following autonomous system
\begin{equation} \left( \begin{array}{c} \widehat{R} \\ \widehat{S} \\ \end{array} \right)_t = \left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu+f'(1)+\varepsilon_1 \\ \end{array} \right) \left( \begin{array}{c} \widehat{R} \\ \widehat{S} \\ \end{array} \right)+\left( \begin{array}{c} 0 \\ h(\widehat{S}) \\ \end{array} \right) \end{equation} | (4.20) |
with the initial data
\begin{equation} \widehat{R}(0)\geq\overline{R}(z, 0), \widehat{S}(0)\geq\overline{S}(z, 0), \forall z\in\mathbb{R}, \end{equation} | (4.21) |
we know that the solution (\widehat{R}, \widehat{S})(t) of (4.20) is an upper solution of system (4.18).
Next, for the convergence of (R, S)(z, t) , it is sufficient to prove that (\widehat{R}, \widehat{S})(t) converges to (0, 0) as t tends to \infty . A direct calculation shows that the eigenvalues \widehat{\lambda}_1 and \widehat{\lambda}_2 of the Jacobian matrix of system (4.20) at the fixed point (0, 0) are less than zero. Therefore, (0, 0) is a stable node. From the phase plane analysis, any manifold in \widehat{R}\widehat{S} -space for any initial value (\widehat{R}, \widehat{S})(0) in region [0, \frac{\mu}{\sigma}]\times[0, 1] converges to the origin (0, 0) . Then, as t\rightarrow \infty , we define
\begin{equation} (\widehat{R}, \widehat{S}) = \widehat{k}_1(\widehat{C}_1, \widehat{C}_2)e^{\widehat{\lambda}_1 t} \end{equation} | (4.22) |
with \widehat{k}_1 > 0 and (\widehat{C}_1, \widehat{C}_2)^T is the eigenvector of the Jacobian matrix respect to \widehat{\lambda}_1 . We can choose \widehat{k}_1 large enough and \widetilde{\lambda}_1 = \min\{\eta_1, -\widehat{\lambda}_1\} to be used that we have
\begin{equation} (R, S)(z_0, t)\leq k_1(\zeta_1, \zeta_2)e^{-\eta_1t}\leq \widehat{k}_1(\zeta_1, \zeta_2) e^{-\widetilde{\lambda}_1t} \end{equation} | (4.23) |
at the boundary z = z_0 . As a result, by comparison on the domain (-\infty, z_0]\times[0, \infty) , see Lemma 3.2 in [36], for all (z, t)\in(-\infty, z_0]\times\mathbb{R}^+ ,
\begin{equation} (R, S)(z, t)\leq \widehat{k}_1(\zeta_1, \zeta_2)e^{-\widetilde{\lambda}_1t}, \end{equation} | (4.24) |
then (R, S)(z, t) converges to (0, 0) for z\in(-\infty, z_0] .
Lemma 4.2. Under the conditions given in Theorem 2.2, (U^-, V^-)(z, t) converges to (\overline{U}, \overline{V})(z) .
Proof. Let
\begin{equation} X(z, t) = \overline{U}(z)-U^-(z, t), Y(z, t) = \overline{V}(z)-V^-(z, t), \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+ \end{equation} | (4.25) |
to satisfy the initial values
\begin{equation} X(z, 0) = \overline{U}(z)-U_0^-(z), Y(z, 0) = \overline{V}(z)-V_0^-(z). \end{equation} | (4.26) |
Similar discussion as lemma 4.1, \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+ , we have
\begin{equation} (0, 0)\leq(X, Y)(z, t)\leq\left(\frac{\mu}{\sigma}, 1\right), \end{equation} | (4.27) |
and
\begin{equation} \left( \begin{array}{c} X \\ Y \\ \end{array} \right)_t = D\left( \begin{array}{c} X \\ Y \\ \end{array} \right)_{zz}+C\left( \begin{array}{c} X \\ Y \\ \end{array} \right)_z+B_{1}\left( \begin{array}{c} X \\ Y \\ \end{array} \right)+\left( \begin{array}{cc} 0 \\ f(V^{-}+Y)-f(V^{-})\\ \end{array} \right), \end{equation} | (4.28) |
where the second-order matrices C and D are consistent with those in (3.3) and
\begin{equation} B_{1} = \left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu \\ \end{array} \right). \end{equation} | (4.29) |
Now we also consider the convergence for the following two cases with respect to z .
Case 1. ( z > z_0 ). By using the fact X < \overline{U} and Y < \overline{V} , we can verify that \exists\eta_2 > 0 and k_2\geq e^{\beta(z-z_0)} \mathop{\max}\limits_{z\in\mathbb{R}}\left\{\frac{X(z, 0)}{\zeta_1}, \frac{Y(z, 0)}{\zeta_2}\right\} bring
\begin{equation} (X, Y)(z, t)\leq k_2(\zeta_1, \zeta_2)e^{-\eta_2t} \end{equation} | (4.30) |
for all (z, t)\in\mathbb{R}\times\mathbb{R}^+ .
Case 2. ( z\leq z_0 ). Define (\widehat{X}, \widehat{Y})(z, t) as the solution of the following autonomous system
\begin{equation} \left( \begin{array}{c} \widehat{X} \\ \widehat{Y} \\ \end{array} \right)_t = \left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu+f'(1)+\varepsilon_1 \\ \end{array} \right) \left( \begin{array}{c} \widehat{X} \\ \widehat{Y} \\ \end{array} \right)+\left( \begin{array}{c} 0 \\ h(\widehat{Y}) \\ \end{array} \right) \end{equation} | (4.31) |
with the initial data
\begin{equation} \widehat{X}(0)\geq\overline{X}(z, 0), \widehat{Y}(0)\geq\overline{Y}(z, 0), \forall z\in\mathbb{R}. \end{equation} | (4.32) |
Then (\widehat{X}, \widehat{Y}) is an upper solution to the system
\begin{equation} \left( \begin{array}{c} X \\ Y \\ \end{array} \right)_t = D\left( \begin{array}{c} X \\ Y \\ \end{array} \right)_{zz}+C\left( \begin{array}{c} X \\ Y \\ \end{array} \right)_z+\left( \begin{array}{cc} -\sigma & \mu \\ \sigma & -\mu \\ \end{array} \right) \left( \begin{array}{c} X \\ Y \\ \end{array} \right)+\left( \begin{array}{c} 0 \\ f(\overline{V})-f(\overline{V}-Y) \\ \end{array} \right). \end{equation} | (4.33) |
Phase plane analysis shows that for any initial values in region [0, \frac{\mu}{\sigma}]\times[0, 1] , (\widehat{X}, \widehat{Y})(t) always converges to \left(\frac{\mu}{\sigma}, 1\right) . Similar to the proof of case 2 in Lemma 4.1, for some positive numbers \widehat{k}_2 and \widetilde{\lambda}_2 , we obtain
\begin{equation} (X, Y)(z, t)\leq \widehat{k}_2(\zeta_1, \zeta_2)e^{-\widehat{\lambda}_2t}, \; \; \forall(z, t)\in(-\infty, z_0]\times\mathbb{R}^+. \end{equation} | (4.34) |
Now we will use the squeezing theorem to obtain the conclusion of the Theorem (2.2).
By the inequalities (4.4), for any (z, t)\in\mathbb{R}\times\mathbb{R}^+ , it gives
\begin{equation} \begin{aligned} & |X(z, t)| \leq|U(z, t)-\overline{U}(z)|\leq|R(z, t)|, \\ & |Y(z, t)| \leq|V(z, t)-\overline{V}(z)|\leq|S(z, t)|. \end{aligned} \end{equation} | (4.35) |
From Lemmas 4.1 and 4.2 and the squeezing theorem, \exists k > 0, \eta > 0 so that \forall(z, t)\in\mathbb{R}\times\mathbb{R}^+ ,
\begin{equation} \begin{aligned} & |U(z, t)-\overline{U}(z)|\leq k e^{-\eta t}, \\ & |V(z, t)-\overline{V}(z)|\leq k e^{-\eta t}. \end{aligned} \end{equation} | (4.36) |
To sum up, Theorem 2 is proved completely.
Y. Tang and C. Pan: Methodology; H. Wang and C. Pan: Validation; Y. Tang and C. Pan: Formal analysis; C. Pan: Writing-original draft preparation; Y. Tang and H. Wang: Writing-review and editing; C.Pan and H. Wang: Supervision; C. Pan: Project administration; C. Pan: Funding acquisition. All authors have read and agreed to the published version of the manuscript.
This work is supported by the National Natural Science Foundation of China grant (No.11801263).
The authors declare no conflict of interest.
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