Nonstandard Quasi-monotonicity: An Application to the Wave Existence in a Neutral KPP–Fisher Equation

Abstract

We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction–diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP–Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone wavefronts.

Appendix

1.1 On the Upper and Lower Solutions

In Definition 2 of super- and sub-solutions, their first derivatives are allowed to have a jump discontinuity whenever the sign of jump is non-positive (for super-solutions) or non-negative (for sub-solutions). Actually, even if the number of such discontinuities does not matter, in practice it suffices to use only one point. Considering $C^2$-smooth (on ${\mathbb {R}}$) functions in Definition 2, we obtain the definition of the upper and lower solutions. Since in general it is rather difficult to find good initial approximations $\phi _\pm$ for the waves, the advantage of working with less demanding super- and sub-solutions instead of upper and lower solutions becomes quite evident. The main result of this section is well known in several particular cases, cf. [15, Lemma 2.5], [28], [5, Lemma 15]. However, the proofs of it given in the mentioned papers are not suitable to cope with the general situation described in Theorem 3. Instead, here we are using the mollification technique.

Lemma 14

Let $C^\infty$-smooth function $k_\delta : {\mathbb {R}}\rightarrow {\mathbb {R}}_+$ have compact support contained in $[-\delta ,\delta ]$ and satisfy $\int _Rk_\delta (s)ds=1$. Suppose that $\phi _+, \phi _-$ are super- and sub-solutions given in Definition 2. Then

$$\begin{aligned} \psi _\pm (t) = \int _{-\delta }^{\delta }\phi _\pm (t-s)k_\delta (s)ds, \quad t \in {\mathbb {R}}, \end{aligned}$$

are $C^\infty$-smooth functions such that

$$\begin{aligned} g_+(t,\delta ):= & {} D\psi _+''(t) -c\psi '_+(t) + \int _{-\delta }^{\delta }k_\delta (s)F(\phi _+)(t-s)ds \le 0, \ t \in {\mathbb {R}}; \end{aligned}$$

(17)

$$\begin{aligned} g_-(t,\delta ):= & {} D\psi _-''(t) -c\psi '_-(t) + \int _{-\delta }^{\delta }k_\delta (s)F(\phi _-)(t-s)ds \ge 0, \ t \in {\mathbb {R}}. \end{aligned}$$

(18)

In addition, $\psi _\pm (t)$ possess another properties of respective $\phi _\pm (t)$ listed in Definition 2. Furthermore, under assumptions of Theorem 3, we have that $\phi _- \le A\phi _-$, $A\phi _+ \le \phi _+$, where $A(\phi ):= -{\mathcal {I}}[F(\phi )+L\phi ]$.

Proof

Clearly,

$$\begin{aligned} \psi _\pm (t) = \int _{t-\delta }^{t+\delta }\phi _\pm (s) k_\delta (t-s)ds \end{aligned}$$

are $C^\infty$-smooth and have the same asymptotic, monotonicity and sign properties as $\phi _\pm (t)$. Next, for all $t \not \in (t_1-\delta , t_1+\delta )$, it is immediate to calculate the first and the second derivatives

$$\begin{aligned} \psi ^{(j)}_\pm (t) = \int _{-\delta }^{\delta }\phi ^{(j)}_\pm (t-s)k_\delta (s)ds, \ j =1,2. \end{aligned}$$

Now, if $t \in (t_1-\delta , t_1+\delta )$, we can use the representation

$$\begin{aligned} \psi _\pm (t) = \int _{t-\delta }^{t_1}\phi _\pm (s) k_\delta (t-s)ds+ \int _{t_1}^{t+\delta }\phi _\pm (s) k_\delta (t-s)ds \end{aligned}$$

to find, after integrating by parts, that

$$\begin{aligned} \psi '_\pm (t)= & {} \int _{-\delta }^{\delta }\phi '_\pm (t-s)k_\delta (s)ds,\\ \psi ''_\pm (t)= & {} \int _{-\delta }^{\delta }\phi ''_\pm (t-s)k_\delta (s)ds + k_\delta (t-t_1)\left[ \phi _\pm '(t_1+)-\phi _\pm '(t_1-) \right] \end{aligned}$$

(in the latter formula, we can set formally $\phi ''_\pm (t_1) =0$).

Next, after multiplying differential inequalities of Definition 2 evaluated in $t-s$ by $k_\delta (s)$ and integrating them between $-\delta$ and $+\delta$ with respect to s, we easily obtain (17) and (18) in view of the assumed sign restrictions on $\phi _\pm '(t_1+)-\phi _\pm '(t_1-)$.

Finally, we have that

$$\begin{aligned} \psi _+(t)= & {} - \int _{{\mathbb {R}}}N(t-s) \left[ -g_+(s,\delta ) + L\psi _+(s)+ \int _{-\delta }^{\delta }k_\delta (v)F(\phi _+)(s-v)dv \right] ds \\\ge & {} -\int _{{\mathbb {R}}}N(t-s) \left[ L\psi _+(s)+ \int _{-\delta }^{\delta }k_\delta (v)F(\phi _+)(s-v)dv \right] ds. \end{aligned}$$

By p-continuity of L and the Lebesgue’s dominated convergence theorem, after taking limit, as $\delta \rightarrow 0+$, in the latter inequality, we find that $\phi _+ \ge A\phi _+$. The proof of inequality $\phi _- \le A\phi _-$ is similar and therefore it is omitted. $\square$

1.2 Curves Defined by the Characteristic Equations

Lemma 15

Assume that $b \in (0,1)$ is a fixed number, $c, \tau > 0$. Then there exist finite $\tau (b)>0$ and a strictly decreasing continuous function $c_\#=c_\#(\tau )>0,\ \tau >\tau (b),$ satisfying $c_\#(\tau (b)-)=+\infty , \ c_\#(+\infty ) =0$ and such that $\chi _1(z)$ has exactly two negative zeros $\mu _2(c) \le \mu _1(c)$ (counting multiplicity) if and only if either $\tau \in (0,\tau (b)]$ or $\tau > \tau (b)$ and $c \le c_\#(\tau )$. These zeros coincide only if $c=c_\#(\tau )$. Function $\tau (b)$ is strictly decreasing on (0, 1), with $\tau (0+) = 1/e$ and $\tau (1-)= 0$.

Proof

It is easy to find that $\chi _1'''(x)\not =0$ for all real $x \not = \ln b/(c\tau )$ and that $\chi _1(x)$ has exactly one non-negative real zero (which is simple, in addition). Thus $\chi _1(z)$ can have at most two negative zeros. After introducing the change of variable $\lambda = cz, \ \epsilon = c^{-2}$, equation $\chi _1(z) =0$ takes the form

$$\begin{aligned} \epsilon z^2 -z = \frac{1}{e^{z\tau }-b}. \end{aligned}$$

(19)

It shows clearly that for each $\tau > 0$ and $b\in (0,1)$ there exists a unique non-negative real number $\epsilon _\#$ such that (19) has negative solutions if and only $\epsilon \ge \epsilon _\#$. Thus $\epsilon _\# =0$ if and only if equation

$$\begin{aligned} -z = \frac{1}{e^{z\tau }-b}, \ z <0. \end{aligned}$$

(20)

has negative solutions. Since the right-hand side of (20) is an increasing function of $\tau$, we deduce that there exists some positive $\tau (b)$ such that (20) has negative roots for all $\tau \in (0,\tau (b)]$ and does not have real roots if $\tau > \tau (b)$. In other words, $\epsilon _\#=0$ if and only if $\tau \in (0,\tau (b)]$. Actually, the above arguments imply all above mentioned properties of $c_\#(\tau )$, see also (23) below. These arguments also shows that $\tau =\tau (b)$ can be determined as a unique postive number for which Eq. (20) has a negative double root $z_0$. Thus

$$\begin{aligned} -z_0 = \frac{1}{e^{z_0\tau }-b}, \quad -1 = \frac{-\tau e^{z_0\tau }}{(e^{z_0\tau }-b)^2}, \end{aligned}$$

(21)

from which

$$\begin{aligned} 1=z_0^2\tau e^{z_0\tau },\,\,\, z_0 b -\frac{1}{z_0\tau } =1, \quad \hbox {so that,} \quad z_0\tau = -2(1+\sqrt{1+4b/\tau })^{-1}=:-\sigma , \ \sigma \in (0,1). \end{aligned}$$

Hence, we have obtained a parametric solution of (21): $\tau = \sigma ^2e^{-\sigma }, \ b = (1-\sigma )e^{-\sigma },$$\sigma \in (0,1)$. Note that $\tau '(b) = -\sigma <0$, other properties of $\tau (b)$ are also obvious. $\square$

Lemma 16

The curves $c=c_*(\tau )$ and $c=c_\#(\tau )$, $\tau > \tau (b)$, have exactly one intersection point $(\tau _0, c_0)$ where $c_*'(\tau _0) >c_\#'(\tau _0)$.

Proof

For $c=c_*(\tau )$ [respectively, $c=c_\#(\tau )$] the characteristic function $\chi _0(z)$ [respectively, $\chi _1(z)$] has positive multiple zeros $\lambda _2(\tau ) = \lambda _1(\tau )$ [respectively, negative zeros $\mu _1(\tau )=\mu _2(\tau )$]. Set $\epsilon (\tau ) = c_*^{-2}(\tau ), \ z(\tau ) = \lambda _2(\tau )c_*(\tau )$, then $\epsilon (\tau ), z(\tau ),\tau$ satisfy the system

$$\begin{aligned} F(z(\tau ),\epsilon (\tau ),\tau ) =0, \quad F_z(z(\tau ),\epsilon (\tau ),\tau ) =0, \quad \hbox {where} \quad F(z,\epsilon ,\tau ) = \epsilon z^2-z + \frac{1}{1-be^{-z\tau }}. \end{aligned}$$

Thus, after differentiating the first relation with respect to $\tau$, we can find that

$$\begin{aligned} c_*'(\tau ) = -\frac{c_*(\tau )}{\tau } + \frac{c^2_*(\tau )}{2\lambda _2(\tau )\tau } <0, \ \tau >0. \end{aligned}$$

(22)

Here we use that obvious fact that $\lambda _2(\tau ) > c_*(\tau )/2$ for $\tau >0$. Similarly,

$$\begin{aligned} c_\#'(\tau ) = -\frac{c_\#(\tau )}{\tau } + \frac{c^2_\#(\tau )}{2\mu _2(\tau )\tau } <0. \end{aligned}$$

(23)

(Curiously, these differential relations coincide with equations in [6, A.3: proof of Lemma 1.3, p. 66] derived in a different situation). It is obvious that due to the asymptotic properties of $c=c_*(\tau )$ and $c=c_\#(\tau )$ described in Lemmas 4 and 15, the graphs of these functions have at least one intersection at some point $(\tau _0, c_0)$. Since $\mu _2(\tau _0)< 0 < \lambda _2(\tau _0)$, differential relations (22) and (23) yield $c_*'(\tau _0) >c_\#'(\tau _0)$. This implies the uniqueness of the intersection point $\tau _0$. $\square$