Utilizing fixed point approach to investigate piecewise equations with non-singular type derivative

1.   Introduction

In this paper, we mainly focus on the following stream-population model^[1]

with the initial data

Here, $u(x, t)$, $v(x, t)$ are the population densities of neuston and benthon respectively; $\sigma$ represents the per capita drift rate at which the species returns to the benthic population; $\mu$ is the per capita rate at which individuals in the benthic population enter the drift; $\alpha$ is the speed of the flow; $d$, $\epsilon$ 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 $v\in(0, 1)$. $\sigma, \mu, \alpha, d, \epsilon$ are positive constants and possess the biological meaning^[2,3]. It is worth mentioning that the benthon basically do not move horizontally since $\epsilon\ll1$. Obviously, $(0, 0)$ and $\left(\frac{\mu}{\sigma}, 1\right)$ are the equilibria and further we can obtain that $(0, 0)$ is unstable and $\left(\frac{\mu}{\sigma}, 1\right)$ 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 $\left(\frac{\mu}{\sigma}, 1\right)$, which possesses the wave profile as

where the wave speed $c\geq0$. Substituting (1.3) into (1.1), yields the new wave profile as

with the boundary value condition

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 $(\overline{U}, \overline{V})(x-ct)$ when $t$ is large enough for given initial data $u_{0}(x)$ and $v_{0}(x)$. To begin with, we set

and then (1.4) can be transformed into the partial differential model

subject to

From the above system (1.7) we know that $(\overline{U}, \overline{V})(z)$ 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 $(\overline{U}, \overline{V})(z)$. 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 $(\overline{U}, \overline{V})(z)$ in a special choice on the weighted function space $L_\omega^\infty(\mathbb{R})$ is investigated in section 4.

2.   Preliminaries and main results

First, linearizing the system (1.4) at $(0, 0)$ we obtain

The transformation

with $\lambda > 0$ and $\xi_{i}(i = 1, 2)$ being positive constants, allows us to transform the discussion of the asymptotic behaviors at infinitely in variable $z$ to the following eigenvalue problem

where $\xi = (\xi_{1}, \xi_{2})^{T}$. Letting

then the eigenvalue problem (2.3) is equivalent to $r(\lambda)\xi = A(\lambda)\xi$. Further, we can obtain the eigenvalues

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 $A(\lambda)$ is

where $r_{+}$ is real number and for any $\lambda\in(0, +\infty)$,

Actually, through the classified discussion of the positive and negative of $A_{1}$and $A_{2}$, $r_{+}(\lambda) > 0$ holds true.

Lemma 2.1. (^[1], see Section 2) $r(\lambda)$, defined in (2.7), is a real, continuous, and convex function with respect to $\lambda\in R$. Assume that

where $c_{0}$ is called the minimal wave speed, then we can obtain that the equation $c\lambda = r(\lambda)$ has

● no solution, if $c < c_{0}$;

● a unique solution $\lambda(c_0)$, if $c = c_{0}$;

● two solutions $\lambda_{1}(c)$ and $\lambda_{2}(c)$ with $\lambda_{1}(c) < \lambda_{2}(c)$, if $c > c_{0}$.

Based on the above analysis, we could give the asymptotic behaviors as $z\rightarrow \infty$ with the following lemma.

Lemma 2.2. (^[1], see Section 2) The asymptotic behaviors of the traveling wave solution $(\overline{U}(z), \overline{V}(z))$ (see (2.1)) can be represented as follows:

with $C_1 > 0$ or $C_1 = 0, C_2 > 0$, where the eigenvectors corresponding to eigenvalues $\lambda_{i}(c)(i = 1, 2)$ are

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, $L^p$ are function spaces defined by using a natural generalization of the $p$-norm for finite dimensional vector spaces. The weighted Lebesgue space $L^p_\omega$ with $1\leq p < \infty$ can be expressed by

with the norm

where the weighted function

for some positive constants $z_{0}$ and $\beta$.

With the introduction above, we state our main conclusions for two theorems.

Theorem 1. (Local stability) For any $c > c_0$, the wavefront $(\overline{U}, \overline{V})(z)$ is locally stable in the weighted function space $L_\omega^p$ if $\beta\in(\lambda_1, \lambda_2)$ to be chosen.

Theorem 2. (Global stability) Assume that $c > c_0$, $\lambda_1 < \beta < \lambda_2$ and the initial data $U(z, 0) = U_0(z)$ and $V(z, 0) = V_0(z)$ satisfy

and

Then (1.7) with initial data admits a unique solution $(U, V)(z, t)$ satisfying

Moreover, the inequalities

and

hold for some positive constants $k$ and $\eta$.

3.   Proof of Theorem 1

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

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:

where

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

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:

where $\Phi = (\phi_{1}, \phi_{2})^{T}$, $D$ is defined in (3.3) and

Since $H(z)$ and $J(z)$ are bounded real matrix functions and $\omega$, $\omega'$, $\omega''$ exist as $z\rightarrow \pm\infty$, we define

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

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

Therefore, the equation $det(-\theta^2 D+i\theta H_++J_+-\chi I) = 0$ has two solutions, i.e.

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

By Euler's formula, we obtain the real parts of the eigenvalues $\chi_{1, 2}$ as follows:

Obviously, $Re(\chi_{1}) > Re(\chi_{2})$. By a simple calculation, one can obtain that

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

Similar to Case 1, solving the equation $det(-\theta^2 D+i\theta H_-+J_–\chi I) = 0$, we obtain

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

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

Let

with $\lambda > 0$ and $\xi_{3}$, $\xi_{4}$ being positive constants. Substituting (3.18) into (3.17), it follows that

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:

with $C_3 > 0$ or $C_3 = 0, C_4 > 0$.

To associate with (3.2), we consider the following system

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:

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

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

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

and $r_{p_{2}}^{-}$, $r_{p_{2}}^{+}$, $r_{p_{4}}^{-}$ and $r_{p_{4}}^{+}$ are the roots of

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.

4.   Proof of Theorem 2

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

respectively; namely

It holds that

By comparison principle, we have

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

which satisfies the initial values

By (4.2) and (4.4), we have

Through (1.4) and (4.3), the following inequality

holds for the matrices $C$ and $D$ defined in (3.3), and

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

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

where $A_1(\beta)$ and $A_2(\beta)$ are defined in (2.4).

Further, we choose

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,

which has the eigenvalue

The associated eigenvectors can be found by direct calculation as follows:

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

Thus, we can choose $\eta_1\leq-\overline{\lambda}_1$ such that

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

then by comparison principle on unbounded domain, $\forall(z, t)\in\mathbb{R}\times\mathbb{R}^+$, we obtain

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

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

for some given enough small positive $\varepsilon_1$ and $\varepsilon_1\ll\mu-f'(1)$.

From the following autonomous system

with the initial data

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

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

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 $(z, t)\in(-\infty, z_0]\times\mathbb{R}^+$,

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

to satisfy the initial values

Similar discussion as lemma 4.1, $\forall(z, t)\in\mathbb{R}\times\mathbb{R}^+$, we have

and

where the second-order matrices $C$ and $D$ are consistent with those in (3.3) and

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

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

with the initial data

Then $(\widehat{X}, \widehat{Y})$ is an upper solution to the system

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

□

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

From Lemmas 4.1 and 4.2 and the squeezing theorem, $\exists k > 0, \eta > 0$ so that $\forall(z, t)\in\mathbb{R}\times\mathbb{R}^+$,

To sum up, Theorem 2 is proved completely.

Author contributions

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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China grant (No.11801263).

Conflict of interest

The authors declare no conflict of interest.