Propagation Phenomena in Periodic Patchy Landscapes with Interface Conditions

Abstract

This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis. We first establish the well-posedness for the Cauchy problem. Next, we give existence and uniqueness results for the positive steady state and we analyze the long-time behavior of the solutions to the evolution problem. Afterwards, based on dynamical systems methods, we investigate the spreading properties and the existence of pulsating traveling waves in the positive and negative directions. It is shown that the asymptotic spreading speed, $c^*$, exists and coincides with the minimal wave speed of pulsating traveling waves in positive and negative directions. In particular, we give a variational formula for $c^*$ by using the principal eigenvalues of certain linear periodic eigenvalue problems.

Comparison Principles

In this appendix, we prove comparison results for the problem (2.9)–(2.10), as well as for a class of more general non-periodic versions of (2.9)–(2.10), and for the patchy model in an interval $(a,b)\subset \mathbb {R}$ composed of finitely many patches, say $I_i$ for $i=1,\ldots ,n$. For the latter, which we first deal with, the landscape (a, b) can be either bounded or unbounded. Set $-\infty \le a=x_0<x_1<\cdots <x_n=b\le +\infty$ and $I_i=(x_{i-1},x_i)$ for $i=1,\ldots ,n$. Since the results will be used in the present paper and in the future work [22], we state them in more generality to cover different applications. We consider a one-dimensional parabolic operator

$$\begin{aligned} {\mathcal {L}}u:=u_t-d(x)u_{xx}-c(t,x)u_x-F(x,u),~~\text {for}~t>0\hbox { and }x\in (a,b)\backslash \{x_1,\ldots ,x_{n-1}\}=\bigcup _{i=1}^nI_i, \end{aligned}$$

with interface conditions

$$\begin{aligned} u(t,x_i^-)=u(t,x_i^+)\hbox { and }u_x(t,x_i^-)=\sigma _iu_x(t,x_i^+),~~\hbox {for }t>0\hbox { and }i=1,\ldots ,n-1. \end{aligned}$$

(A.1)

If a or b is finite, we impose Dirichlet-type boundary conditions:

$$\begin{aligned} u(t,a)=\varphi ^-(t)~~\text {or}~~u(t,b)=\varphi ^+(t),~~\text {for}~t\ge 0, \end{aligned}$$

(A.2)

where $\varphi ^\pm :[0,+\infty )\rightarrow {\mathbb {R}}$ are given continuous functions. Here, the function $x\mapsto d(x)$ is assumed to be constant and positive in each patch, i.e., $d|_{I_i}=d_i>0$ for some constant $d_i$,⁷ the function c is assumed to be continuous and bounded in $(0,T_0)\times \cup _{i=1}^nI_i$ for every $T_0\in (0,+\infty )$, the $\sigma _i$’s are given positive real numbers, and, for each $1\le i\le n$, $F(x,s)=f_i(s)$ for $(x,s)\in I_i\times {\mathbb {R}}$, with $f_i\in C^1({\mathbb {R}})$.

We first give the definition of super- and subsolutions of ${\mathcal {L}}u=0$ associated with the interface and boundary conditions (A.1)–(A.2).

Definition A.1

For $T\in (0,+\infty ]$, we say that a continuous function $\overline{u}:[0,T)\times \overline{(a,b)}\rightarrow {\mathbb {R}}$,⁸ which is assumed to be bounded in $[0,T_0]\times \overline{(a,b)}$ for every $T_0\in (0,T)$, is a supersolution for the problem ${\mathcal {L}}u=0$ with interface and boundary conditions (A.1)–(A.2), if $\overline{u}|_{(0,T)\times \overline{I_i}}\in C^{1;2}_{t;x}((0,T)\times \overline{I_i})$ satisfies ${\mathcal {L}}\overline{u}|_{(0,T)\times I_i}\ge 0$ in the classical sense for each $1\le i\le n$, and if

$$\begin{aligned} \overline{u}_x(t,x_i^-)\ge \sigma _i \overline{u}_x(t,x_i^+),~~\text {for}~t\in (0,T)\text { and}~i=1,\ldots ,n-1, \end{aligned}$$

and

$$\begin{aligned} \overline{u}(t,a)\ge \varphi ^-(t)~\text { or }~\overline{u}(t,b)\ge \varphi ^+(t),~~\text {for}~t\in [0,T), \end{aligned}$$

provided that a or b is finite. A subsolution can be defined in a similar way with all the inequality signs above reversed.

The first result of the appendix is a comparison principle between super- and subsolutions when the interval (a, b) is bounded.

Proposition A.2

(Comparison principle in bounded intervals). Assume that $-\infty<a<b<+\infty$. For $T\in (0,+\infty ]$, let $\overline{u}$ and $\underline{u}$ be, respectively, a super- and a subsolution in $[0,T)\times [a,b]$ of ${\mathcal {L}}u=0$ with (A.1)–(A.2), and assume that $\overline{u}(0,\cdot )\ge \underline{u}(0,\cdot )$ in [a, b]. Then, $\overline{u}\ge \underline{u}$ in $[0,T)\times [a,b]$ and, if $\overline{u}(0,\cdot )\not \equiv \underline{u}(0,\cdot )$, then $\overline{u}>\underline{u}$ in $(0,T)\times (a,b)$.

Proof

Fix any $T_0\in (0,T)$ and set

$$\begin{aligned} M:=\max \left( \Vert \overline{u}\Vert _{L^\infty ([0,T_0]\times [a,b])},\Vert \underline{u}\Vert _{L^\infty ([0,T_0]\times [a,b])}\right) \ \hbox { and }\ \mu :=\max \limits _{1\le i\le n}\Vert f'_i\Vert _{L^\infty ([-M,M])}\nonumber \\ \end{aligned}$$

(A.3)

(notice that M and $\mu$ are nonnegative real numbers owing to the assumptions on $\overline{u}$, $\underline{u}$ and $f_i$). Define

$$\begin{aligned} w(t,x):=\left( \overline{u}(t,x)-\underline{u}(t,x)\right) \,e^{-\mu t}\ \hbox { for }~(t,x)\in [0,T_0]\times [a,b]. \end{aligned}$$

The function w is continuous in $[0,T_0]\times [a,b]$, with restriction in $(0,T_0]\times \overline{I_i}$ of class $C^{1;2}_{t;x}((0,T_0]\times \overline{I_i})$ for each $1\le i\le n$, and we see from the mean value theorem that w satisfies

$$\begin{aligned} {\mathcal {N}}w:=w_t-d(x)w_{xx}-c(t,x)w_x+\big (\mu -F_s(x,\eta (t,x))\big )w\ge 0,~\text {for}~(t,x)\in (0,T_0]\!\times \!\bigcup \limits _{i=1}^n I_i, \end{aligned}$$

(A.4)

where $\eta (t,x)$ is an intermediate value between $\overline{u}(t,x)$ and $\underline{u}(t,x)$ (hence, $|\eta (t,x)|\le M$ and $\mu -F_s(x,\eta (t,x))\ge 0$). Moreover, there holds

$$\begin{aligned} w_x(t,x_i^-)\ge \sigma _i w_x(t,x_i^+),~~\text {for}~t\in (0,T_0]~\text {and}~i=1,\ldots ,n-1, \end{aligned}$$

(A.5)

together with $w(0,x)=\overline{u}(0,x)-\underline{u}(0,x)\ge 0$ for all $x\in [a,b]$, $w(t,a)\ge 0$ and $w(t,b)\ge 0$ for all $t\in [0,T_0]$.

Consider now an arbitrary $\varepsilon >0$ and let us introduce the auxiliary function z defined by

$$\begin{aligned} z(t,x):=w(t,x)+\varepsilon (t+1)\ \hbox { for}~ (t,x)\in [0,T_0]\times [a,b]. \end{aligned}$$

The function z has at least the same regularity as w, and $z>0$ in $\{0\}\times [a,b]$ and in $[0,T_0]\times \{a,b\}$. Moreover,

$$\begin{aligned} {\mathcal {N}}z={\mathcal {N}}w+\varepsilon + \big (\mu -F_s(x,\eta (t,x))\big )\varepsilon (t+1)\ge \varepsilon >0,~~\text {for}~(t,x)\in (0,T_0]\times \bigcup \limits _{i=1}^n I_i,\nonumber \\ \end{aligned}$$

(A.6)

with

$$\begin{aligned} z_x(t,x_i^-)\ge \sigma _i z_x(t,x_i^+),~~\text {for}~t\in (0,T_0]~\text {and}~i=1,\ldots ,n-1. \end{aligned}$$

(A.7)

We claim that $z(t,x)>0$ for all $(t,x)\in [0,T_0]\times [a,b]$. Assume not. Then, by continuity, there is a point $(t_0,y_0)\in (0,T_0]\times (a,b)$ such that $z(t_0,y_0)=\min _{[0,t_0]\times [a,b]}z=0$. We first assume that $y_0\in I_i$ for some $1\le i\le n$. Since $z_t(t_0,y_0)\le 0$, $z_x(t_0,y_0)=0$ and $z_{xx}(t_0,y_0)\ge 0$, we see that

$$\begin{aligned} {\mathcal {N}}z(t_0,y_0)=z_t(t_0,y_0)-d_iz_{xx}(t_0,y_0)+c(t_0,y_0)z_x(t_0,y_0)+\big (\mu -f_i'(\eta (t_0,y_0))\big )z(t_0,y_0)\le 0,\nonumber \\ \end{aligned}$$

(A.8)

which is impossible by (A.6). Thus, necessarily, we can assume without loss of generality that $y_0=x_i$ for some $1\le i\le n-1$ and that $z>0$ in $[0,t_0]\times \cup _{i=1}^nI_i$. Then, the Hopf lemma yields

$$\begin{aligned} z_x(t_0,x_i^-)<0\ \hbox { and }\ z_x(t_1,x_i^+)>0, \end{aligned}$$

which contradicts (A.7). Consequently, $z>0$ in $[0,T_0]\times [a,b]$. Since $\varepsilon >0$ was arbitrarily chosen, we obtain that $w\ge 0$ in $[0,T_0]\times [a,b]$, which immediately implies $\overline{u}\ge \underline{u}$ in $[0,T_0]\times [a,b]$, and then in $[0,T)\times [a,b]$ since $T_0\in (0,T)$ was arbitrary.

Let us now further assume that $\overline{u}(0,\cdot )\not \equiv \underline{u}(0,\cdot )$ in [a, b], hence by continuity $\overline{u}(0,\cdot )>\underline{u}(0,\cdot )$ in some non-empty open subinterval of (a, b) which has a non-empty intersection with $I_i$, for some $1\le i\le n$. Since we already know from the previous paragraph that $\overline{u}\ge \underline{u}$ in $[0,T)\times [a,b]$, it follows from the interior strong parabolic maximum principle that $\overline{u}>\underline{u}$ in $(0,T)\times I_i$. If the interval (a, b) reduces to a single patch (that is, $n=1$), then we are done. Otherwise, either $x_{i-1}$ or $x_i$ belongs to the open interval (a, b). Let us consider the case when $x_i\in (a,b)$ (hence, $i\le n-1$). We now claim that $\overline{u}(t,x_i)>\underline{u}(t,x_i)$ for all $t\in (0,T)$. Indeed, otherwise, there is a time $t_0\in (0,T)$ such that $\overline{u}(t_0,x_i)=\underline{u}(t_0,x_i)$, and the Hopf lemma then implies that

$$\begin{aligned} \overline{u}_x(t_0,x_i^-)<\underline{u}_x(t_0,x_i^-). \end{aligned}$$

But $\overline{u}_x(t_0,x_i^+)\ge \underline{u}_x(t_0,x_i^+)$ since $\overline{u}\ge \underline{u}$ in $[0,T)\times \overline{I_{i+1}}$ and $\overline{u}(t_0,x_i)=\underline{u}(t_0,x_i)$. One finally gets a contradiction with the assumptions on the spatial derivatives of the super- and subsolutions $\overline{u}$ and $\underline{u}$ at $x_i^\pm$. Therefore, $\overline{u}(t,x_i)>\underline{u}(t,x_i)$ for all $t\in (0,T)$. By continuity and by applying the strong interior parabolic maximum principle in $(0,T)\times I_{i+1}$, we infer that $\overline{u}>\underline{u}$ in $(0,T)\times I_{i+1}$. By an immediate induction, going from one patch to the adjacent one in the left or right directions, we get that $\overline{u}>\underline{u}$ in $(0,T)\times (a,b)$. The proof of Proposition A.2 is thereby complete. $\square$

Then we prove in Proposition A.3 the comparison principle when $(a,b)=\mathbb {R}$, still in the case of a finite number of interfaces (the case when the domain is of the form $(a,+\infty )$ with $a\in {\mathbb {R}}$, or $(-\infty ,b)$ with $b\in {\mathbb {R}}$, can be handled by a combination and a slight modification of the proofs of Propositions A.2 and A.3 ).

Proposition A.3

(Comparison principle in $\mathbb {R}$ with finitely many interfaces). For $T\in (0,+\infty ]$, let $\overline{u}$ and $\underline{u}$ be, respectively, a super- and a subsolution in $[0,T)\times {\mathbb {R}}$ of ${\mathcal {L}}u=0$ with (A.1), and assume that $\overline{u}(0,\cdot )\ge \underline{u}(0,\cdot )$ in ${\mathbb {R}}$. Then, $\overline{u}\ge \underline{u}$ in $[0,T)\times {\mathbb {R}}$ and, if $\overline{u}(0,\cdot )\not \equiv \underline{u}(0,\cdot )$, then $\overline{u}>\underline{u}$ in $(0,T)\times {\mathbb {R}}$.

Proof

Fix any $T_0\in (0,T)$ and define the nonnegative real numbers M and $\mu$ as in (A.3) with this time ${\mathbb {R}}$ instead of [a, b] in the definition of M. Denote $w(t,x):=(\overline{u}(t,x)-\underline{u}(t,x))e^{-\mu t}$ for $(t,x)\in [0,T_0]\times {\mathbb {R}}$. The function w is continuous and bounded in $[0,T_0]\times {\mathbb {R}}$, with restriction in $(0,T_0]\times \overline{I_i}$ of class $C^{1;2}_{t;x}((0,T_0]\times \overline{I_i})$ for each $1\le i\le n$ (notice that, here, $I_1=(-\infty ,x_1)$ and $I_n=(x_{n-1},+\infty )$ are unbounded), and w still satisfies (A.4)–(A.5), together with $w(0,\cdot )=\overline{u}(0,\cdot )-\underline{u}(0,\cdot )\ge 0$ in ${\mathbb {R}}$. Set now $R=\max _{1\le i\le n-1}|x_i|+1>0$, and let $\varrho :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a nonnegative $C^2$ function with bounded first and second order derivatives, and satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \varrho =0\hbox { in }[-R,R],\ \ \lim _{x\rightarrow +\infty }\varrho (x)=+\infty ,\\ \displaystyle \Big (\max _{1\le i\le n}d_i\Big )\times \Vert \varrho ''\Vert _{L^\infty ({\mathbb {R}})}+\Vert c\Vert _{L^\infty ((0,T_0]\times \cup _{i=1}^nI_i)}\times \Vert \varrho '\Vert _{L^\infty ({\mathbb {R}})}\le \frac{1}{2}.\end{array}\right. \end{aligned}$$

Let us consider an arbitrary $\varepsilon >0$, and introduce an auxiliary function z defined by

$$\begin{aligned} z(t,x):=w(t,x)+\varepsilon (\varrho (|x|)+t+1)\ \hbox { for}~ (t,x)\in [0,T_0]\times {\mathbb {R}}. \end{aligned}$$

The function z has at least the same regularity as w, while $z(0,x)\ge \varepsilon >0$ for all $x\in {\mathbb {R}}$ and $z(t,x)\rightarrow +\infty$ as $|x|\rightarrow +\infty$ uniformly in $t\in [0,T_0]$. Moreover,

$$\begin{aligned} {\mathcal {N}}z\ge {\mathcal {N}}w+\varepsilon -\varepsilon d(x)\varrho ''(|x|)-\varepsilon |c(t,x)\varrho '(|x|)|+ \big (\mu -F_s(x,\eta (t,x))\big )\varepsilon (\varrho (|x|)+t+1)\ge \frac{\varepsilon }{2}>0 \end{aligned}$$

for $(t,x)\in (0,T_0]\times \bigcup \limits _{i=1}^n I_i$, and (A.7) still holds from (A.5), the definition of R and the choice of $\varrho$. We claim that $z(t,x)>0$ for all $(t,x)\in [0,T_0]\times {\mathbb {R}}$. Assume not. Then, by continuity and the above properties of z, there is a point $(t_0,y_0)\in (0,T_0]\times {\mathbb {R}}$ such that $z(t_0,y_0)=\min _{[0,t_0]\times {\mathbb {R}}}z=0$. If $y_0\in I_i$ for some $1\le i\le n$, then we see as in (A.8) that ${\mathcal {N}}z(t_0,y_0)\le 0$, which is impossible. Thus, we can assume without loss of generality that $y_0=x_i$ for some $1\le i\le n-1$ and that $z>0$ in $[0,t_0]\times \cup _{i=1}^nI_i$. Then, the Hopf lemma yields $z_x(t_0,x_i^-)<0$ and $z_x(t_1,x_i^+)>0$, contradicting (A.7). Consequently, $z>0$ in $[0,T_0]\times {\mathbb {R}}$. Hence, by passing to the limit as $\varepsilon \rightarrow 0^+$, we infer that $w\ge 0$ in $[0,T_0]\times {\mathbb {R}}$, that is, $\overline{u}\ge \underline{u}$ in $[0,T_0]\times {\mathbb {R}}$, and then $\overline{u}\ge \underline{u}$ in $[0,T)\times {\mathbb {R}}$ owing to the arbitrariness of $T_0\in (0,T)$.

Lastly, if one further assumes that $\overline{u}(0,\cdot )\not \equiv \underline{u}(0,\cdot )$, then the proof of the strict inequality $\overline{u}>\underline{u}$ in $(0,T)\times \mathbb {R}$ follows similar lines as in the proof of the preceding proposition. $\square$

The last statement is a comparison principle for a class, more general than (2.9)–(2.10), of non-periodic problems involving countably many interfaces. Namely, we are given a countable set $S=\{x_i:i\in {\mathbb {Z}}\}\subset {\mathbb {R}}$ with

$$\begin{aligned} \delta :=\inf _{i\in {\mathbb {Z}}}\,(x_{i+1}-x_i)>0, \end{aligned}$$

(A.9)

and we consider the problem

$$\begin{aligned} \left\{ \begin{array}{rcll} u_t-d(x)u_{xx}-c(t,x)u_x &{} = &{} F(x,u), &{} t>0,\ x\in \mathbb {R}\!\setminus \!S,\\ u(t,x_i^-) &{} = &{} u(t,x_i^+), &{} t> 0,\ i\in {\mathbb {Z}},\\ u_x(t,x_i^-) &{} = &{} \sigma _i u_x(t,x_i^+), &{} t> 0,\ i\in {\mathbb {Z}}.\end{array}\right. \end{aligned}$$

(A.10)

We assume that the function $x\mapsto d(x)$ is equal to a positive constant $d_i$ in each interval $(x_i,x_{i+1})$, and that $\sup _{i\in {\mathbb {Z}}}d_i\!<\!+\infty$. The function c is assumed to be continuous and bounded in $(0,T_0)\times ({\mathbb {R}}\!\setminus \!S)$ for every $T_0\in (0,+\infty )$, the $\sigma _i$’s are given positive real numbers, and there are $C^1({\mathbb {R}})$ functions $(f_i)_{i\in {\mathbb {Z}}}$ such that $F(x,s)=f_i(s)$ for every $(x,s)\in (x_i,x_{i+1})\times {\mathbb {R}}$ and $i\in {\mathbb {Z}}$, with $\sup _{i\in {\mathbb {Z}}}\Vert f'_i\Vert _{L^\infty ([-L,L])}\!<\!+\infty$ for every $L>0$.

For $T\in (0,+\infty ]$, we say that a continuous function $\overline{u}:[0,T)\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, which is assumed to be bounded in $[0,T_0]\times {\mathbb {R}}$ for every $T_0\in (0,T)$, is a supersolution of (A.10) in $[0,T)\times {\mathbb {R}}$, if, for every $i\in {\mathbb {Z}}$, the function $\overline{u}|_{(0,T)\times [x_i,x_{i+1}]}$ is of class $C^{1;2}_{t;x}((0,T)\times [x_i,x_{i+1}])$ and satisfies $u_t(t,x)-d_iu_{xx}(t,x)-c(t,x)u_x(t,x)\ge F(x,u(t,x))$ for every $(t,x)\in (0,T)\times (x_i,x_{i+1})$, and if $\overline{u}_x(t,x_i^-)\ge \sigma _i \overline{u}_x(t,x_i^+)$ for every $i\in {\mathbb {Z}}$ and $t\in (0,T)$. A subsolution is defined similarly with all the inequality signs reversed.

The following result provides a comparison between sub- and supersolutions of (A.10) with ordered initial conditions, thus yielding the uniqueness of solutions for given initial conditions.

Proposition A.4

(Comparison principle for problems of type (A.10)) For $T\in (0,+\infty ]$, let $\overline{u}$ and $\underline{u}$ be, respectively, a super- and a subsolution of (A.10) in $[0,T)\times {\mathbb {R}}$ with $\overline{u}(0,\cdot )\ge \underline{u}(0,\cdot )$ in ${\mathbb {R}}$. Then, $\overline{u}\ge \underline{u}$ in $[0,T)\times {\mathbb {R}}$, and, if $\overline{u}(0,\cdot )\not \equiv \underline{u}(0,\cdot )$, then $\overline{u}>\underline{u}$ in $(0,T)\times {\mathbb {R}}$.

Proof

Fix any $T_0\in (0,T)$. Define

$$\begin{aligned} M\!:=\!\max \!\big (\Vert \overline{u}\Vert _{L^\infty ([0,T_0]\times {\mathbb {R}})},\Vert \underline{u}\Vert _{L^\infty ([0,T_0]\times {\mathbb {R}})}\big )\hbox { and }\mu \!:=\!\sup _{i\in {\mathbb {Z}}}\Vert f'_i\Vert _{L^\infty ([-M,M])}\!=\!\!\sup _{x\in {\mathbb {R}}\setminus S,\,|s|\le M}\!|F_s(x,s)|, \end{aligned}$$

which are two nonnegative real numbers. Denote $w(t,x):=(\overline{u}(t,x)-\underline{u}(t,x))e^{-\mu t}$ for $(t,x)\in [0,T_0]\times {\mathbb {R}}$. The function w is continuous and bounded in $[0,T_0]\times {\mathbb {R}}$, and it still satisfies inequalities similar to (A.4) (with, here, $\cup _{i=1}^nI_i$ replaced by ${\mathbb {R}}\setminus S$), together with $w_x(t,x_i^-)\ge \sigma _iw_x(t,x_i^+)$ for every $i\in {\mathbb {Z}}$ and $t\in (0,T_0]$. Furthermore, $w(0,\cdot )=\overline{u}(0,\cdot )-\underline{u}(0,\cdot )\ge 0$ in ${\mathbb {R}}$.

Let now $(\rho _m)_{m\in {\mathbb {N}}}$ be a family of nonnegative $C^\infty ({\mathbb {R}})$ mollifiers with unit mass and such that each function $\rho _m$ has a support included in $[-1/m,1/m]$. Remember that $\delta >0$ is defined in (A.9). With $\mathbbm {1}_E$ denoting the characteristic function of a set E, and $\star$ being the convolution product, we then define

$$\begin{aligned} F:=\mathop {\bigcup }_{i\in {\mathbb {Z}}}\,\left( x_i+\frac{3\delta }{10},x_{i+1}-\frac{3\delta }{10}\right) \end{aligned}$$

and

$$\begin{aligned} \phi :=\rho _m\star \left( -\mathbbm {1}_{F\cap (-\infty ,-1/2)}+\mathbbm {1}_{F\cap (1/2,+\infty )}\right) , \end{aligned}$$

with a certain m large enough so that the $C^\infty ({\mathbb {R}})$ function $\phi$ satisfies $\phi \le 0$ in $(-\infty ,0]$, $\phi \ge 0$ in $[0,+\infty )$, $\phi =-1$ in $[x_i+2\delta /5,x_{i+1}-2\delta /5]\cap (-\infty ,-1]$, $\phi =1$ in $[x_i+2\delta /5,x_{i+1}-2\delta /5]\cap [1,+\infty )$ and $\phi =0$ in $[x_i-\delta /5,x_i+\delta /5]$, for all $i\in {\mathbb {Z}}$. Notice that $-1\le \phi \le 1$ in ${\mathbb {R}}$ and that $\phi '$ is bounded in ${\mathbb {R}}$. Let us then define

$$\begin{aligned} \varrho (x):=\int _0^x\phi (y)\,\mathrm {d}y \end{aligned}$$

for $x\in {\mathbb {R}}$. The $C^\infty ({\mathbb {R}})$ function $\varrho$ is nonnegative, it has bounded first and second order derivatives, and $\varrho (x)\rightarrow +\infty$ as $x\rightarrow \pm \infty$. There is then a positive real number $\kappa >0$ such that

$$\begin{aligned} \left( \sup _{i\in {\mathbb {Z}}}d_i\right) \times \Vert \kappa \varrho ''\Vert _{L^\infty ({\mathbb {R}})} +\Vert c\Vert _{L^\infty ((0,T_0]\times ({\mathbb {R}}\setminus S))}\times \Vert \kappa \varrho '\Vert _{L^\infty ({\mathbb {R}})}\le \frac{1}{2}. \end{aligned}$$

Let us then consider an arbitrary $\varepsilon >0$, and introduce an auxiliary function z defined by

$$\begin{aligned} z(t,x):=w(t,x)+\varepsilon (\kappa \varrho (x)+t+1)\ \hbox { for}~ (t,x)\in [0,T_0]\times {\mathbb {R}}. \end{aligned}$$

The function z is continuous in $[0,T_0]\times {\mathbb {R}}$, and it satisfies

$$\begin{aligned} z(0,x)\ge \varepsilon >0\hbox { for all}~ x\in {\mathbb {R}}\hbox { , and}~ z(t,x)\rightarrow +\infty ~ \hbox { as}~ |x|\rightarrow +\infty ~\hbox { uniformly in}~ t\in [0,T_0],\nonumber \\ \end{aligned}$$

(A.11)

since w is bounded in $[0,T_0]\times {\mathbb {R}}$ and $\varrho (\pm \infty )=+\infty$. Moreover, with the same notations as in (A.4), one has

$$\begin{aligned} {\mathcal {N}}z(t,x)= & {} \underbrace{{\mathcal {N}}w(t,x)}_{\ge 0}+\varepsilon -\underbrace{(\varepsilon d(x)\kappa \varrho ''(x)\!+\!\varepsilon c(t,x)\kappa \varrho '(x))}_{\le \varepsilon /2}+\underbrace{(\mu \!-\!F_s(x,\eta (t,x)))}_{\ge 0}\underbrace{\varepsilon (\varrho (x)+t+1)}_{\ge 0}\\\ge & {} \displaystyle \frac{\varepsilon }{2}>0 \end{aligned}$$

for all $(t,x)\in (0,T_0]\times ({\mathbb {R}}\!\setminus \!S)$, while $z_x(t,x_i^-)\ge \sigma _iz_x(t,x_i^+)$ for all $t\in (0,T_0]$ and $i\in {\mathbb {Z}}$ (since w satisfies these inequalities and $\varrho '(x_i)=\phi (x_i)=0$ for each $i\in {\mathbb {Z}}$).

We claim that $z>0$ in $[0,T_0]\times {\mathbb {R}}$. Assume not. Then, by continuity and (A.11), there is $(t_0,y_0)\in (0,T_0]\times {\mathbb {R}}$ such that $z(t_0,y_0)=\min _{[0,t_0]\times {\mathbb {R}}}z=0$. If $y_0\in (x_i,x_{i+1})$ for some $i\in {\mathbb {Z}}$, then we see as in (A.8) that ${\mathcal {N}}z(t_0,y_0)\le 0$, which is impossible. Thus, one can assume without loss of generality that $y_0=x_i$ for some $i\in {\mathbb {Z}}$ and that $z>0$ in $[0,t_0]\times ({\mathbb {R}}\setminus S)$, whence $z_x(t_0,x_i^-)<0$ and $z_x(t_0,x_i^+)>0$ from the Hopf lemma, which is again impossible. As a consequence, $z>0$ in $[0,T_0]\times {\mathbb {R}}$, hence $w\ge 0$ in $[0,T_0]\times {\mathbb {R}}$ due to the arbitrariness of $\varepsilon >0$, and finally $\overline{u}\ge \underline{u}$ in $[0,T)\times {\mathbb {R}}$ due to the arbitrariness of $T_0\in (0,T)$.

Lastly, if one further assumes that $\overline{u}(0,\cdot )\not \equiv \underline{u}(0,\cdot )$ in ${\mathbb {R}}$, then one concludes as in the proof of Proposition A.2 that $\overline{u}>\underline{u}$ in $(0,T)\times \mathbb {R}$. $\square$