Entire solutions of monotone bistable reaction–diffusion systems in $\pmb {\mathbb {R}}^N$

Abstract

This paper is concerned with entire solutions of monotone bistable reaction–diffusion systems in $\mathbb {R}^N$. We first show that there is a new entire solution for bistable reaction–diffusion systems which behaves as three moving planar fronts as time goes to $-\,\infty$ and as a V-shaped traveling front as time goes to $\infty$. Then we address the speed of the interfaces corresponding to the entire solution by appealing to the super-sub solutions technique. Finally, we apply our results to two important models in biology.

Appendix

In this section, we give a sufficient condition to ensure that the assumption (T) holds. We further list the following assumption

(A5) :

$\varvec{F}$ is of class $C^{1+\beta }$ on the open domain $\Omega \supset [\varvec{0},\varvec{1}]$, where $\beta \in (0,1)$. The off-diagonal elements of the Jacobian matrices $\varvec{F}'(\varvec{0})$ and $\varvec{F}'(\varvec{1})$ are positive.

We will give exact exponential behaviors of the traveling fronts $\varvec{\Phi }(\xi )$ of (1.1) at $\xi =\pm \infty$ under the assumptions (A1)–(A5) and then show that (T) holds. Here we emphasize that we do not concern the existence of planar traveling fronts of (1.1) and we always assume that (1.1) admits a unique traveling front $\varvec{\Phi }(\varvec{x}\cdot \varvec{e}-ct)=(\Phi _1(\varvec{x}\cdot \varvec{e}-ct),\ldots , \Phi _m(\varvec{x}\cdot \varvec{e}-ct))$ satisfies (1.2). Of course, as mentioned in Sect. 1, the existence of bistable planar traveling fronts of (1.1) have been widely studied, in particular, see [8, 44, 45].

To show the main results of this section, we first make some preparations. Let $\varvec{\Phi }(\varvec{x}\cdot \varvec{e}-ct)$ be the unique traveling front of (1.1) satisfying (1.2), namely, $\varvec{\Phi }(\varvec{x}\cdot \varvec{e}-ct)$ satisfies

$$\begin{aligned}&\varvec{D}\varvec{\Phi }''(\xi )+c\varvec{\Phi }'(\xi )+\varvec{F}(\varvec{\Phi }(\xi ))=\varvec{0},\quad \lim _{\xi \rightarrow \infty }\varvec{\Phi }(\xi )=\varvec{0},\quad \lim _{\xi \rightarrow -\infty }\varvec{\Phi }(\xi )=\varvec{1},\nonumber \\&\quad \text {and}\quad \varvec{\Phi }'(\xi )<\varvec{0}\ \text {for all }\xi \in \mathbb {R}. \end{aligned}$$

(5.1)

Let $\varvec{w}_0$ be an equilibrium of $\varvec{F}$, namely, $\varvec{F}(\varvec{w}_0)=\varvec{0}$. Setting $\varvec{Y} =\varvec{\Phi }'$ and $\varvec{Z}=\varvec{\Phi }''$, then from (5.1) we have

$$\begin{aligned} \left( \begin{array}{c} \varvec{\Phi }\\ \varvec{Y} \\ \end{array} \right) '=\varvec{M}\left( \begin{array}{c} \varvec{\Phi }\\ \varvec{Y} \\ \end{array} \right) +\left( \begin{array}{c} \varvec{0} \\ \varvec{D}^{-1}(\varvec{F}'(\varvec{w}_0)-\varvec{F}(\varvec{\Phi })) \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \left( \begin{array}{c} \varvec{Y} \\ \varvec{Z} \\ \end{array} \right) '=\varvec{M}\left( \begin{array}{c} \varvec{Y}\\ \varvec{Z} \\ \end{array} \right) +\left( \begin{array}{c} \varvec{0} \\ \varvec{D}^{-1}(\varvec{F}'(\varvec{w}_0)-\varvec{F}'(\varvec{\Phi }))\varvec{Y} \\ \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} \varvec{M}=\left( \begin{array}{cc} \varvec{0} &{} \varvec{I} \\ -\varvec{D}^{-1}\varvec{F}'(\varvec{w}_0) &{} -c \varvec{D}^{-1} \\ \end{array} \right) . \end{aligned}$$

Consider the equation

$$\begin{aligned} (\lambda ^2\varvec{D}+c\lambda \varvec{I}+\varvec{F}'(\varvec{w}_0))\varvec{q}=0, \end{aligned}$$

(5.2)

where $\lambda \in \mathbb {C}$ and $\varvec{q}\in \mathbb {C}^m{\setminus } \{0\}$. According to [7, 9], we call these $\lambda$ and $\varvec{q}$ satisfying (5.2) as TWP-eigenvalues and eigenvectors (eigenvalues and eigenvectors of the traveling wave problem) at $\varvec{w}_0$. A TWP-eigenvalue $\lambda$ with eigenvector $\varvec{q}$, is said to be stable (unstable) monotone if $\lambda \in \mathbb {R}$ is negative (positive) and all the components of $\varvec{q}$ are non-zero of the same sign. An equilibrium $\varvec{w}_0$ is called to be stable if all the eigenvalues of $\varvec{F}'(\varvec{w}_0)$ are in the open left-half complex plane and unstable if there is an eigenvalue in the open right-half plane. Now we list four lemmas. The first two lemmas are borrowed from [9] and the last two come from [45].

Lemma 5.1

[9, Theorem A.1]

(i) :

If $\varvec{w}_0$ is a stable equilibrium, then for every $c\in \mathbb {R}$, there is exactly one stable monotone and one unstable monotone TWP-eigenvalue at $(\varvec{w}_0; \varvec{0})$, say $\lambda ^-(c)<0<\lambda ^+(c)$.

(ii) :

If $\varvec{w}_0$ is an unstable equilibrium, then there exist ${\tilde{c}}_0 < c_0\in \mathbb {R}$ such that for $c < c_0$ $(c > {\tilde{c}}_0)$, there are no stable (unstable) monotone TWP-eigenvalues at $(\varvec{w}_0, \varvec{0})$, when $c = c_0$ $(c = {\tilde{c}}_0)$, there is exactly one stable (unstable) monotone TWP-eigenvalue, and for $c > c_0$ $(c < {\tilde{c}}_0)$, there are precisely two stable (unstable) monotone TWP-eigenvalues, $\lambda ^-(c)<\lambda ^+(c)<0$ $(0< \lambda ^-(c)<\lambda ^+(c))$.

Lemma 5.2

[9, Theorem A.2] Suppose that $\varvec{w}_0$ is either a stable equilibrium or that $\varvec{w}_0$ is unstable and $c > c_0$ (so there are two monotone eigenvalues at $(\varvec{w}_0,\varvec{0})$, $\lambda ^-(c)<\lambda ^+(c)$). Then

(i) :

for $\lambda \in \mathbb {R}$, $\mu _{PF} (\lambda ^2\varvec{D}+c\lambda \varvec{I}+\varvec{F}'(\varvec{w}_0)) < 0$ (namely, the real, simple Perron–Frobenius eigenvalue of $\lambda ^2\varvec{D}+c\lambda \varvec{I}+\varvec{F}'(\varvec{w}_0)$ is negative)$\iff$ $\lambda ^-(c)<\lambda <\lambda ^+(c)$;

(ii) :

if $\lambda$ is a TWP-eigenvalue with $\lambda ^-(c)\le \text {Real}\ \lambda \le \lambda ^+(c)$, then $\lambda \in \{\lambda ^-(c),\lambda ^+(c)\}$;

(iii) :

$\lambda ^-(c)$ and $\lambda ^+(c)$ are simple TWP-eigenvalues, namely, $\lambda ^-(c)$ and $\lambda ^+(c)$ are all simple eigenvalue of $\varvec{M}$;

(iv) :

when $\varvec{w}_0$ is stable $( \lambda ^-(c)<0<\lambda ^+(c))$, $\lambda ^-(c)$ and $\lambda ^+(c)$ are both decreasing functions of c; when $\varvec{w}_0$ is unstable and $c > c_0$ $( \lambda ^-(c)<\lambda ^+(c)<0)$, $\lambda ^-(c)$ and $\lambda ^+(c)$ are decreasing and increasing functions of c, respectively.

Lemma 5.3

[45, Page 163, Lemma 2.4] Let $\varvec{F}(\varvec{w}_0) = \varvec{0}$ and the off-diagonal elements of $\varvec{F}'(\varvec{w}_0)$ are positive. If there exists a solution $\varvec{\Phi }(\xi )$ of system (1.2) tending towards $\varvec{w}_0$ as $\xi \rightarrow \infty$ $(\xi \rightarrow -\infty )$ and such that $\varvec{\Phi }(\xi ) > \varvec{w}_0$ for all sufficiently large $\xi$ $(-\xi )$, then there exist a number $\lambda \le 0$ $(\lambda \ge 0)$ and a positive vector $\varvec{q}$, such that (5.2) holds. Furthermore, if the maximum eigenvalue of the matrix $\varvec{F}'(\varvec{w}_0)$ is not equal to 0, then, obviously, $\lambda \ne 0$.

Lemma 5.4

([45, Page 239, Lemma 4.1]) Assume that a real matrix T has nonnegative off-diagonal elements and a negative principal (i.e., with the maximal real part) eigenvalue. Suppose, further, that S is a complex diagonal matrix whose elements have nonpositive real parts. Then all the eigenvalues of the matrix $T + S$ have negative real parts.

Following Lemmas 5.1 and 5.2, we know that there exist a unique stable monotone TWP-eigenvalue at the equilibrium $\varvec{0}$ and a unique unstable monotone TWP-eigenvalue at the equilibrium $\varvec{1}$. Denote them by $\lambda _0$ and $\mu _1$, respectively. Clearly, $\lambda _0<0$ and $\mu _1>0$. Now we are in the position to present the main result of this section, which is concerned with the explicit exponential behaviors of $\varvec{\Phi }$, $\varvec{\Phi }'$ and $\varvec{\Phi }''$ at $\pm \infty$. The main proofs are very similar to those of Crooks [9, Lemma 3.2].

Theorem 5.5

Suppose that (A1)–(A5) hold. Let $\varvec{\Phi }(\xi )$ be such that (5.1). Let $\lambda _0<0$ and $\mu _1>0$ be the unique stable/unstable monotone TWP-eigenvalue at $\varvec{0}$ and $\varvec{1}$, respectively. Then there are two vectors $\varvec{\alpha }^-\gg \varvec{0}$ and $\varvec{\alpha }^+\gg \varvec{0}$ such that

$$\begin{aligned}&e^{-\lambda _0 \xi }\varvec{\Phi }(\xi )\rightarrow \varvec{\alpha }^+ \quad \text {as } \xi \rightarrow \infty , \end{aligned}$$

(5.3)

$$\begin{aligned}&e^{-\mu _1 \xi }(\varvec{1}-\varvec{\Phi }(\xi ))\rightarrow \varvec{\alpha }^- \quad \text {as } \xi \rightarrow -\infty , \end{aligned}$$

(5.4)

$$\begin{aligned}&e^{-\lambda _0 \xi }\varvec{\Phi }'(\xi )\rightarrow \lambda _0\varvec{\alpha }^+ \quad \text {as }\xi \rightarrow \infty , \end{aligned}$$

(5.5)

$$\begin{aligned}&e^{-\mu _1 \xi }\varvec{\Phi }'(\xi )\rightarrow -\mu _1\varvec{\alpha }^- \quad \text {as } \xi \rightarrow -\infty , \end{aligned}$$

(5.6)

$$\begin{aligned}&e^{-\lambda _0 \xi }\varvec{\Phi }''(\xi )\rightarrow \lambda _0^2\varvec{\alpha }^+ \quad \text {as }\xi \rightarrow \infty , \end{aligned}$$

(5.7)

$$\begin{aligned}&e^{-\mu _1 \xi }\varvec{\Phi }''(\xi )\rightarrow -\mu _1^2\varvec{\alpha }^- \quad \text {as } \xi \rightarrow -\infty . \end{aligned}$$

(5.8)

Proof

It suffices to show (5.3), (5.5) and (5.7), since (5.4), (5.6) and (5.8) can be proved in a similar way.

We first claim that there are no TWP-eigenvalues at $\varvec{0}$ with zero real part. By the assumptions (A2) and (A5) and the Perron–Frobenius theorem, the principle eigenvalue of $\varvec{F}'(\varvec{0})$ is a negative real number. Therefore, 0 can not be a TWP-eigenvalues at $\varvec{0}$. Thus, we suppose on the contrary that $\lambda =i\mu$ is a TWP-eigenvalues at $\varvec{0}$, where $\mu \in \mathbb {R}\backslash \{0\}$. It is clear that the matrix $-\mu ^2\varvec{D}+\varvec{F}'(\varvec{0})$ has positive off-diagonal elements and a negative principal eigenvalue. By Lemma 5.4, one gets that all the eigenvalues of the matrix $-\mu ^2\varvec{D}+\varvec{F}'(\varvec{0})+ic\mu \varvec{I}$ have negative real parts. Clearly, this contradicts the fact that 0 is an eigenvalue of the matrix $-\mu ^2\varvec{D}+\varvec{F}'(\varvec{0})+ic\mu \varvec{I}$.

Following such a claim and the proof of [5, Page 330, Theorem 4.1], we have that $\varvec{\Phi }(\xi )$ and $\varvec{\Phi }'(\xi )$ tend to $\varvec{0}$ exponentially as $\xi$ goes to the positive infinity. As a result of [5, Page 338, Theorem 4.5] we have

$$\begin{aligned} \varvec{\Phi }(\xi )=\sum _{i=1}^k Q_i(\xi )e^{\lambda _i\xi }+O(e^{(b-\delta )\xi }),\quad \xi >0 \end{aligned}$$

(5.9)

where $\delta > 0$, $k\in \mathbb N$, $Q_i(\xi ) \in \mathbb C^N \backslash \{\varvec{0}\}$ are polynomials in $\xi$, and $\lambda _i$ are TWP-eigenvalues at $\varvec{0}$ with $\text {Real}\ \lambda _k =\text {Real}\ \lambda _{k-1} = \cdots = \text {Real}\ \lambda _1 = b <0$. Moreover, by the fact that $\varvec{\Phi }(\xi )> \varvec{0}$ and $\varvec{\Phi }(\xi )\rightarrow \varvec{0}$ as $\xi \rightarrow \infty$ and the proof of Lemma 5.3 we can conclude that the right-hand side of (5.9) is dominated by a term of the form $\xi ^{j-1}e^{\hat{\lambda }\xi }\varvec{\alpha }^+$ for large $\xi$, where $\mathbb {R}\ni \hat{\lambda }= b < 0$ is a TWP-eigenvalue at $\varvec{0}$ with TWP-eigenvector $\varvec{\alpha }^+ \ (\in \mathbb {R}^m) \gg \varvec{0}$ and $j\in \mathbb N$ is the algebraic multiplicity of $\hat{\lambda }$ as an eigenvalue of the $2m \times 2m$ stability matrix $\varvec{M}$ at $\varvec{0}$. Therefore, $\hat{\lambda }$ is a stable monotone TWP-eigenvalue, and whence $\hat{\lambda }=\lambda _0$ from Lemma 5.1. Furthermore, we obtain from Lemma 5.2 that $m=1$ and $k=1$. As a consequence, we arrive at

$$\begin{aligned} \varvec{\Phi }(\xi )=e^{\lambda _0 \xi }\varvec{\alpha }^++O(e^{(\lambda _0-\delta )\xi }),\quad \xi >0. \end{aligned}$$

Thus, we have completed the proof of (5.3).

It remains to show (5.5) and (5.7). Recall that $\varvec{D}\varvec{\Phi }''+c\varvec{\Phi }'+\varvec{F}(\varvec{\Phi })=\varvec{0}$ for any $\xi \in \mathbb {R}$ and $\varvec{\Phi }(\xi ), \varvec{\Phi }'(\xi )$ tend to $\varvec{0}$ exponentially as $\xi \rightarrow \infty$. Integrating the system $\varvec{D}\varvec{\Phi }''+c\varvec{\Phi }'+\varvec{F}(\varvec{\Phi })=\varvec{0}$ from $\xi$ to $\infty$, we get

$$\begin{aligned} \varvec{\Phi }'(\xi )=\varvec{D}^{-1}\left( -c\varvec{\Phi }(\xi )+\int _\xi ^\infty \varvec{F}(\varvec{\Phi })d\xi \right) . \end{aligned}$$

It then follows from the L’Hôpital’s rule that

$$\begin{aligned} \varvec{\Phi }'(\xi )e^{-\lambda _0\xi }= & {} \varvec{D}^{-1}\left( -c\varvec{\Phi }(\xi )+\int _\xi ^\infty \varvec{F}(\varvec{\Phi }(\eta ))d\eta \right) e^{-\lambda _0\xi }\\\rightarrow & {} \varvec{D}^{-1}\left( -c\varvec{I} -\frac{\varvec{F}'(\varvec{0})}{\lambda _0}\right) \varvec{\alpha }^+\quad \text {as }\xi \rightarrow \infty . \end{aligned}$$

Since $\lambda _0$ and $\varvec{\alpha }^+$ satisfy $(\lambda _0^2\varvec{D}+c\lambda _0 \varvec{I}+\varvec{F}'(\varvec{0}))\varvec{\alpha }^+=\varvec{0}$, we have

$$\begin{aligned} \varvec{D}^{-1}\left( -c\varvec{I} -\frac{\varvec{F}'(\varvec{0})}{\lambda _0}\right) \varvec{\alpha }^+=\lambda _0\varvec{\alpha }^+, \end{aligned}$$

which implies that $\lim _{\xi \rightarrow \infty }\varvec{\Phi }'(\xi )e^{-\lambda _0\xi }=\lambda _0\varvec{\alpha }^+$. Again doing the same treatment to the system $\varvec{D}\varvec{\Phi }''+c\varvec{\Phi }'+\varvec{F}(\varvec{\Phi })=\varvec{0}$, we get $\lim _{\xi \rightarrow \infty }\varvec{\Phi }''(\xi )e^{-\lambda _0\xi }=\lambda _0^2\varvec{\alpha }^+$. This completes the proof. $\square$

A direct consequence of Theorem 5.5 is the following corollary.

Corollary 5.6

Suppose that (A1)–(A5) hold. Let $\varvec{\Phi }(\xi )$ satisfy (1.2). Then the assumption (T) holds.

Now we prove Lemma 4.2.

Proof of Lemma 4.2

Define

$$\begin{aligned} f_1(u,v):=f(u)+k_1b_1v-k_2u(b_0-b_1v)\quad \text {and}\quad f_2(u,v):=-k_1 v+\frac{k_2}{b_1}u(b_0-b_1v). \end{aligned}$$

It is easy to check that $\frac{\partial f_1}{\partial v}=k_1b_1+k_2b_1u>0$ and $\frac{\partial f_2}{\partial u}=\frac{k_2}{b_1}(b_0-b_1v)\ge \frac{k_2}{b_1}(b_0-b_1)>0$ for $u,v\in [0,1]$ due to $b_1=k_2b_0/(k_1+k_2)$ and $k_1$, $k_2$, $b_0$ and $b_1$ are positive constants, which implies that system (4.6) satisfies (A4) and (A5). It is also easy to verify that system (4.6) satisfies (A1)–(A3). It then can be inferred from Theorem 5.5 that Lemma 4.2 holds for $\lambda <0$ is the unique stable monotone TWP-eigenvalue at (0, 0) and $\mu >0$ is the unique unstable monotone TWP-eigenvalue at (1, 1), and $(\beta _1,\beta _2)$ and $(\alpha _1,\alpha _2)$ are the TWP-eigenvectors associated with $\lambda$ and $\mu$, respectively. This completes the proof of Lemma 4.2. $\square$