Free boundary models for mosquito range movement driven by climate warming

Abstract

As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and warming have been increasingly recognized as major ones. In this paper, we make use of three diffusive logistic models with free boundary in one space dimension to explore the impact of climate warming on the movement of mosquito range. First, a general model incorporating temperature change with location and time is introduced. In order to gain insights of the model, a simplified version of the model with the change of temperature depending only on location is analyzed theoretically, for which the dynamical behavior is completely determined and presented. The general model can be modified into a more realistic one of seasonal succession type, to take into account of the seasonal changes of mosquito movements during each year, where the general model applies only for the time period of the warm seasons of the year, and during the cold season, the mosquito range is fixed and the population is assumed to be in a hibernating status. For both the general model and the seasonal succession model, our numerical simulations indicate that the long-time dynamical behavior is qualitatively similar to the simplified model, and the effect of climate warming on the movement of mosquitoes can be easily captured. Moreover, our analysis reveals that hibernating enhances the chances of survival and successful spreading of the mosquitoes, but it slows down the spreading speed.

Appendix

1.1 Existence and uniqueness

First we present the following local existence and uniqueness result to problem (2.7) by the contraction mapping theorem.

Theorem 5.1

For any given $\gamma$ satisfying $0<\gamma <1$, a(x) satisfying (2.6) and $u_0$ satisfying (2.5), there is a $T>0$ such that problem (2.7) admits a unique solution

$$\begin{aligned} (u, h)\in C^{(1+\gamma )/2, 1+\gamma }(\overline{D}_{T})\times C^{1+\gamma /2}([0,T]); \end{aligned}$$

moreover,

$$\begin{aligned} \Vert u\Vert _{C^{ (1+\gamma )/2, 1+\gamma }(\overline{D}_{T})}+|h\Vert _{C^{1+\gamma /2}([0,T])}\le C, \end{aligned}$$

(5.1)

where $D_{T}=\{(t,x)\in \mathbb {R}^2: t\in [0,T]\, x\in [0, h(t)]\}$, C and T only depend on $\gamma , h_0$, $\Vert a\Vert _{C^{\alpha }([0, 2h_0])}$ and $\Vert u_0\Vert _{C^{2}([0, h_0])}$.

Proof

The proof is similar as that of Theorem 2.1 in Du and Lin (2010). We give a sketch here. First we straighten the free boundary by the following transformation

$$\begin{aligned} y=\frac{h_0x}{h(t)},\ \ u(t, x)=u\left( t, \frac{h(t)}{h_0}y\right) =w(t, y). \end{aligned}$$

Then the free boundary problem (2.7) becomes

$$\begin{aligned} \left\{ \begin{array}{lll} w_{t}-D\frac{h^2_0}{h^2(t)} w_{yy}-\frac{h'(t)}{h(t)}yw_y=w\left[ a(\frac{h(t)}{h_0}y)-b w\right] ,\; &{}t>0, \ 0<y<h_0, \\ w=0,\quad h'(t)=-\mu \frac{h_0}{h(t)}\frac{\partial w}{\partial y}+\alpha a_{-}\left( \frac{h(t)}{h_0}y\right) ,\quad &{}t>0, \ y=h_0,\\ \frac{\partial w}{\partial y}(t, 0)=0,\; &{}t>0,\\ h(0)=h_0, \quad w(0, y)=u_{0}(y),\; &{}0\le y\le h_0. \end{array} \right. \end{aligned}$$

(5.2)

For fixed $t>0$, as long as $h(t)>0$, the above transformation $y=h_0x/h(t)$ is a diffeomorphism from $[0, +\infty )$ onto $[0, +\infty )$. It changes the free boundary $x=h(t)$ to the line $y=h_0$ but at the expense of making the equation more complicated, since now the coefficients in the first equation of (5.2) include the unknown function h(t).

Denote $h^*=-\mu u_0'(h_0)+\alpha a_{-}(h_0)$ and set

$$\begin{aligned} H_T= & {} \Big \{h\in C^1[0,T]: \, h(0)=h_0, \ h'(0)=h^*, \ h^*-1\le h'\le h^*+1\},\\ W_T= & {} \Big \{w\in C([0,T]\times [0, h_0]): \, w(0,y)=u_0(y),\, \Vert w-u_0\Vert _{C([0,T]\times [0, h_0])}\leqslant 1\Big \}. \end{aligned}$$

If $T\le T_1:=\frac{h_0}{2(|h^*|+1)}$ and $h\in H_T$, then $\frac{1}{2}h_0\le h(t)\le \frac{3}{2}h_0$. It is not difficult to see that $\Gamma _T:=W_T\times H_T$ is a complete metric space with the metric

$$\begin{aligned} \mathcal {D}((w_1,h_1),(w_2,h_2))=\Vert w_1-w_2\Vert _{C([0,T]\times [0, h_0])}+ \Vert h'_1-h'_2\Vert _{C([0,T])}, \end{aligned}$$

and that for $h_1, h_2\in H_T$, by virtue of $h_1(0)=h_2(0)=h_0$,

$$\begin{aligned} \Vert h_1-h_2\Vert _{C([0,T])}\le T||h'_1-h'_2||_{C([0, T])}. \end{aligned}$$

(5.3)

Now, we can apply the standard $L^p$ theory and the Sobolev imbedding theorem (Ladyženskaja et al. 1968) to obtain that for any $(w, h)\in \Gamma _T$, the following initial boundary value problem

$$\begin{aligned} \left\{ \begin{array}{lll} {\tilde{w}}_{t}-D\frac{h^2_0}{h^2(t)} {\tilde{w}}_{yy}-\frac{h'(t)}{h(t)}y{\tilde{w}}_y= w\left( a\left( \frac{h(t)}{h_0}y\right) -b w\right) ,\quad \ &{}0<t\le T,\ 0<y<h_0,\\ \tilde{w}_y(t,0)=\tilde{w}(t,h_0)=0,&{}0<t\le T,\\ \tilde{w}(0,y)=u_0(y)\geqslant 0, &{} 0\leqslant y\leqslant h_0 \end{array} \right. \end{aligned}$$

(5.4)

admits a unique solution $\tilde{w}\in C^{(1+\gamma )/2,1+\gamma }([0,T]\times [0, h_0])$ and

$$\begin{aligned} \Vert \tilde{w}\Vert _{C^{(1+\gamma )/2,1+\gamma }([0,T]\times [0, h_0])}\leqslant C\Vert \tilde{w}\Vert _{W^{1, 2}_ p([0,T]\times [0, h_0])}\leqslant C_1, \end{aligned}$$

(5.5)

where $p=\frac{3}{1-\gamma }$, and $C_1$ is a constant depending on $\gamma , h_0$ and $\Vert u_0\Vert _{C^{2}[0, h_0]}$.

From the third equation in (5.2), we can define a function $\tilde{h}(t)$ as follows

$$\begin{aligned} \tilde{h}(t)=h_0+\int ^t_0\left[ -\mu \frac{h_0}{h(\tau )}\frac{\partial {\tilde{w}}}{\partial y}(\tau , h_0)+\alpha a_{-}(h(\tau ))\right] \text {d}\tau , \end{aligned}$$

(5.6)

which implies $\tilde{h}'(t)=-\mu \frac{h_0}{h(t)}\frac{\partial {\tilde{w}}}{\partial y}(t,h_0)+\alpha a_{-}(h(t))$, ${\tilde{h}}(0)=h_0$ and $\tilde{h}'(0)=h^*$. Hence, $\tilde{h}'(t)\in C^{\gamma /2}([0,T])$ with

$$\begin{aligned} \Vert \tilde{h}'(t)\Vert _{C([0,T])}\leqslant & {} C_2:=\mu C_1+\alpha \Vert a(x)\Vert _{C([0,2h_0])}, \end{aligned}$$

(5.7)

$$\begin{aligned} \Vert \tilde{h}'(t)\Vert _{C^{\gamma /2}([0,T])}\leqslant & {} C_3:=C_2+\mu C_1\left( 1+\frac{C_1}{h_0}\right) \nonumber \\&+\alpha \Vert a(x)\Vert _{C^{\gamma /2}([0,2h_0])}(|h^*|+1)^{\gamma /2}, \end{aligned}$$

(5.8)

where we have used the fact that

$$\begin{aligned} \frac{|a_-(h(t_1))-a_-(h(t_2))|}{(t_1-t_2)^{\gamma /2}}\le & {} \frac{|a(h(t_1))-a(h(t_2))|}{(t_1-t_2)^{\gamma /2}}\\= & {} \frac{|a(h(t_1))-a(h(t_2))|}{|h(t_1)-h(t_2)|^{\gamma /2}} \left( \frac{|h(t_1)-h(t_2)|}{t_1-t_2}\right) ^{\gamma /2} \end{aligned}$$

for $t_1>t_2$.

Next, we define a map $\mathcal {F}$: $\Gamma _{T}\rightarrow C([0,T]\times [0,h_0])\times C^1[0,T]$ by

$$\begin{aligned} \mathcal {F}(w(t,y), h(t)) = (\tilde{w}(t,y), \tilde{h}(t)). \end{aligned}$$

It is easy to see that $(w(t,y), h(t))\in \Gamma _T$ is a fixed point of $\mathcal {F}$ if and only if it solves (5.2).

It follows from (5.5), (5.7) and (5.8) that

$$\begin{aligned} ||\tilde{h}'-h^*||_{C([0,T])}\le & {} ||\tilde{h}'|| _{C^{\gamma /2}([0,T])}T^{\gamma /2}\le C_3T^{\gamma /2},\\ ||\tilde{w} -w_0||_{C([0,T]\times [0,h_0])}\le & {} ||\tilde{w}-w_0||_ {C^{(1+\gamma ) /2, 0}([0,T]\times [0,h_0])} T^{(1+\gamma )/2}\le C_1T^{(1+\gamma ) /2}. \end{aligned}$$

Therefore, if we take $T\le \min \{T_1,\ (C_3)^{-2/\gamma },\ C_1^{-2/(1+\gamma )}\}$, then $\mathcal {F}$ maps $\Gamma _T$ into itself.

The rest of the proof is similar as that in Du and Lin (2010). By using the $L^p$ estimates for parabolic equations and Sobolev’s imbedding theorem, we can prove that $\mathcal {F}$ is a contraction mapping over $\Gamma _T$ for $T>0$ sufficiently small. Then the contraction mapping theorem gives that $\mathcal {F}$ has a unique fixed point (w, h) in $\Gamma _T$. By the Schauder estimates again, we have additional regularity and then (w(y, t), h(t)) is a unique local classical solution of problem (5.2), and therefore (u(t, x), h(t)) is a unique local classical solution of problem (2.7). $\square$

Theorem 5.2

The solution of problem (2.7) exists and is unique for all $t\in (0,\infty )$. Moreover, there exist constants $C_4$ and $C_5$ such that

$$\begin{aligned} 0<u(x, t)\le & {} C_4\quad \text{ for } \quad 0\le x<h(t),\; t\in (0, \infty ).\\ \min \{h_0, h_a\}\le & {} h(t)\quad \text{ for } \quad t\in (0, \infty ), \end{aligned}$$

where $h_a$ is the smallest zero of a(x).

Proof

Let (u, h) be a solution to (2.7) defined for $t\in (0,T_0]$ for some $T_0\in (0, +\infty )$. The positivity of u is directly from the strong maximum principle. Also the maximum principle shows that

$$\begin{aligned} u(t,x)\le \max \Big \{\max _{[0, h_0]}u_0(x),\, a(0)/b \Big \}:=C_4\ \text{ for } \, 0\le x<h(t),\; t\in (0, T_0]. \end{aligned}$$

It is easy to check that $u_x(t,h(t))< 0$, and we can also prove that $u_x(t,h(t))\ge -C_6$ for $t\in (0,T_0]$ and some $C_6$ which is independent of $T_0$ by using the similar method as that of Lemma 2.2 in Du and Lin (2010). Therefore,

$$\begin{aligned} \alpha a_*\le \ h'(t)\le \alpha a(0)+\mu C_6:=C_5 \; \text{ for } \; t\in (0, \infty ). \end{aligned}$$

We claim that $h(t)\ge \min \{h_0, h_a\}$. Otherwise, there exists a $t_0>0$ such that $h(t_0)<\min \{h_0, h_a\}$ and $h'(t_0)\le 0$. But the free boundary condition implies that $h'(t)> \alpha a_-(h(t))$ for all $t>0$. We then have $h'(t_0)> \alpha a_-(h(t_0))=0$, which leads to a contradiction.

Since $h(t)\ge \min \{h_0, h_a\}>0$, u and $h'(t)$ are bounded in $(0, T_0]\times [0,h(t))$ by constants independent of $T_0$, the global existence of the solution is guaranteed. Moreover, the above estimates hold for $t\in (0, \infty )$. $\square$

1.2 Comparison principle

Theorem 5.3

(The Comparison Principle) Assume that $\overline{h}\in C^1([0, +\infty ))$, $g\in C([0, \infty ))$, $0\le g(t)<\overline{h}(t)$ for $t\ge 0$, $D=\{(t,x): t>0, g(t)<x<h(t)\}$, $\overline{u}(t,x) \in C^{1,0}(\overline{D} )\cap C^{1,2}(D)$, and

$$\begin{aligned} \left\{ \begin{array}{lll} \overline{u}_t-D\overline{u}_{xx} \ge a(x)\overline{u}-b\overline{u}^2,&{} t>0,\, g(t)<x<\overline{h}(t), \\ \overline{u}(t,x)=0,\, &{}t>0,\, x=\overline{h}(t),\\ \overline{h}(0)\ge h_0, \; \overline{h}'(t)\ge [-\mu \overline{u}_x+\alpha a_{-}(x)]|_{x=\overline{h}(t)}, &{} t>0,\\ \overline{u}(0,x)\ge u_0(x), &{}g(0)\le x\le h_0. \end{array} \right. \end{aligned}$$

If the solution (u, h) to the free boundary problem (2.7) satisfies $u(t, g(t))\le \overline{u}(t, g(t))$ for $t\ge 0$, then

$$\begin{aligned} h(t)\le \overline{h}(t),\quad \; u(t, x)\le \overline{u}(t,x),\quad \forall t\ge 0,\, \forall x\in [g(t), h(t)]. \end{aligned}$$

Proof

Similarly as in the proof of Lemma 3.5 in Du and Lin (2010) or Lemma 2.6 in Du and Lin (2014), we only prove it for the case that $h(0)<\overline{h}(0)$, the result for the general case $h(0)\le \overline{h}(0)$ can be established by approximation.

First under the assumption that $h(0)<\overline{h}(0)$, we claim that $h(t)<\overline{h}(t)$ for all $t\in (0, \infty )$. If our claim does not hold, then we can find a first $T^*>0$ such that $h(t)<\overline{h}(t)$ for $t\in (0, T^*)$, and $h(T^*)=\overline{h}(T^*)$. It follows that

$$\begin{aligned} \overline{h}'(T^*)\le h'(T^*). \end{aligned}$$

(5.9)

We now compare u and $\overline{u}$ over the bounded region

$$\begin{aligned} \Omega _{T^*}:=\{(x,t)\in \mathbb {R}^2: 0<x<h(t),\, 0< t\le T^*\}. \end{aligned}$$

Since $U(t,x):=\overline{u}(t,x)-u(t,x)$ satisfies

$$\begin{aligned} \left\{ \begin{array}{lll} U_{t}-D U_{xx} \ge a(x) U-b(\overline{u}+u)U,\; &{} 0<t\le T^*, \,0<x<h(t),\\ U_x(t,0)=0,\, U(t,h(t))=\overline{u}(t, h(t))>0,&{} 0<t< T^*, \end{array} \right. \end{aligned}$$

using the strong maximum principle and the Hopf boundary lemma yields $U(x, t)>0$ in $\Omega _{T^*}$, and $U_x(T^*, h(T^*))<0$. We then deduce that

$$\begin{aligned} \overline{h}'(T^*)-h'(T^*)= & {} [-\mu \overline{u}_x(T^*, \overline{h}(T^*))+\alpha a_{-}(\overline{h}(T^*))]\\&-[-\mu u_x(T^*, h(T^*))+\alpha a_{-}(h(T^*))]\\= & {} -\mu U_x(T^*, h(T^*))>0. \end{aligned}$$

But this contradicts (5.9). This proves our claim that $h(t)<\overline{h}(t)$ for all $t\in (0, \infty )$. Using again the maximum principle over $\{(t,x): 0<t<T,\, 0\le x\le h(t)\}$ for any $T>0$ we obtain that $u\le \overline{u}$ for $x\in [0, h(t)]$ and $t\in (0, +\infty )$. $\square$

The pair $(\overline{u}(t,x), \overline{h}(t))$ in Theorem 5.3 is usually called an upper solution of (2.7). Similarly, we can define a lower solution by reversing all of the inequalities in the obvious places. Moreover, it is easy to prove that an analogue of Theorem 5.3 for lower solutions holds.

1.3 Proof of Lemma 3.3

We first observe that, if $a(L)\ge 0$ and hence $a_-(L)=0$, then the conclusion of Lemma 3.3 is evident. Indeed, if there exists $t_0>0$ such that $h(t_0)=L$, then

$$\begin{aligned} h'(t_0)=-\mu u_x(t_0, h(t_0))+\alpha a_-(L)=-\mu u_x(t_0, h(t_0))>0. \end{aligned}$$

It follows immediately that $h(t)-L$ can change sign at most once for $t>0$.

Therefore we assume from now on that $a(L)<0$ and so

$$\begin{aligned} \sigma :=\frac{\alpha }{\mu }a_-(L)<0. \end{aligned}$$

For our proof below, we will make use of the following auxiliary problem:

$$\begin{aligned} -DV''=a(x)V-bV^2,\; V>0 \quad \text{ for } x<L;\;\;\; V(L)=0,\; V'(L)=\sigma . \end{aligned}$$

(5.10)

Standard ODE theory for initial value problems guarantees that (5.10) has a unique solution V(x) and one of the following happens:

• (i) $V(x)>0$ for all $x<L$.

• (ii) There exists $l\in [0,L)$ such that $V(l)=0,\; V(x)>0\; \forall x\in (l, L).$

• (iii) There exists $l\in (-\infty , 0)$ such that $V(l)=0,\; V(x)>0\; \forall x\in (l, L).$

In the following, we will prove Lemma 3.3 for cases (i), (ii) and (iii) separately. In every case, our proof will be based on the zero number argument. To this end, we first extend a(x) to $x<0$ as an even function and still donote it by a(x). We similarly extend u(t, x) to $x\in [-h(t), 0)$ as an even function of x, and still denote it by u(t, x). We will consider the number of zeros of the function

$$\begin{aligned} w(t,\cdot ):=u(t,\cdot )-V(\cdot ) \end{aligned}$$

over a certain interval of x where both $u(t,\cdot )$ and $V(\cdot )$ are defined, and examine how this zero number varies as t increases. Note that, for $t>0$ and $x\in (-h(t), h(t))\cap (l, L)$, w satisfies an equation of the form

$$\begin{aligned} w_t-Dw_{xx}=c(t,x)w \end{aligned}$$

for some $L^\infty$ function c(t, x), where we take $l=-\infty$ when case (i) happens.

We first recall a result which follows directly from Theorem D of Angenent (1988) [see Lemma 2.8 in Cai et al. (2014)].

Lemma 5.4

Let $\xi _1(t)<\xi _2(t)$ be two continuous functions for $t\in (t_1, t_2)$. If u(t, x) satisfies

$$\begin{aligned} \left\{ \begin{array}{l} u_t-Du_{xx}=C(t,x)u\quad \text{ for } \quad x\in (\xi _1(t), \xi _2(t)),\; t\in (t_1, t_2),\\ u(t,\xi _1(t))\not =0,\; u(t,\xi _2(t))\not =0\quad \text{ for } \quad t\in (t_1, t_2), \end{array} \right. \end{aligned}$$

for some $L^\infty$ function C(t, x), then for each $t\in (t_1, t_2)$, the number of zeros of $u(t,\cdot )$ in $I(t):=[\xi _1(t),\xi _2(t)]$ is finite. If we denote this number by $Z[\xi _1(t), \xi _2(t)]$, or simply by Z(t), then $t\rightarrow Z(t)$ is nonincreasing in t, and if for some $s\in (t_1, t_2)$ the function $u(s,\cdot )$ has a degenerate zero $x_0\in I(s)$, then

$$\begin{aligned} Z(s_1)>Z(s_2) \; \text{ for }\, s_1, s_2\; \text{ satisfying }\; t_1<s_1<s<s_2<t_2. \end{aligned}$$

1.3.1 Proof of Lemma 3.3 when V(x) falls into case (i)

Denote

$$\begin{aligned} l(t):=-h(t),\; L(t)=\min \{h(t), L\}. \end{aligned}$$

Arguing indirectly we assume that $h(t)-L$ changes sign infinitely many times as $t\rightarrow \infty$. Then we can find $t_1>t_0>0$ such that

$$\begin{aligned} h(t_0)=h(t_1)=L,\; h(t)<L \;\forall t\in (t_0, t_1). \end{aligned}$$

It follows that

$$\begin{aligned} h'(t_1)\ge 0. \end{aligned}$$

(5.11)

Clearly

$$\begin{aligned} w(t, l(t))= & {} -V(-h(t))<0 \text{ for } \text{ all } t>0,\; w(t, L(t))\\= & {} -V(h(t))<0 \text{ for } t\in (t_0, t_1). \end{aligned}$$

Therefore we can apply Lemma 5.4 to conclude that, for $t\in (t_0, t_1)$, the zero number $\mathcal {Z}(t):=Z[l(t), L(t)]$ of $w(t,\cdot )$ over the interval [l(t), L(t)] is finite, and has the properties stated in Lemma 5.4. It follows that there can exist at most finitely many time moments in $(t_0, t_1)$ such that $w(t,\cdot )$ has a degenerate zero in (l(t), L(t)). Therefore we can find ${\tilde{t}}_1\in (t_0, t_1)$ such that $w(t,\cdot )$ has a finite number of nondegenerate zeros for $t\in [{\tilde{t}}_1, t_1)$. This implies that $\mathcal {Z}(t)$ is a constant for $t\in [{\tilde{t}}_1, t_1)$. We first show that this constant is not 0. Otherwise we have $w(t,x)<0$ for $x\in [l(t), L(t)]$ and $t\in [{\tilde{t}}_1, t_1)$. We may then compare (u(t, x), h(t)) with $(\overline{u}(t,x), \overline{h}(t)):=(V(x), L)$ over $t\ge {\tilde{t}}_1$, $x\in [l(t), h(t)]$ to conclude that $h(t)\le L$ and $u(t,x)\le V(x)$ for such t and x. By the strong maximum principle, one further deduces $h(t)<L$ for all $t>{\tilde{t}}_1$, a contradiction to the existence of $t_1$.

We may now assume $m=\mathcal {Z}(t)\ge 1$ for $t\in [{\tilde{t}}_1, t_1)$. By the implicit function theorem, these m nondegenerate zeros of $w(t,\cdot )$ can be expressed as $C^1$ functions $\gamma _i(t)$ for $t\in [{\tilde{t}}_1, t_1)$, $i=1,\ldots , m$, with

$$\begin{aligned} l(t)<\gamma _1(t)<\cdots<\gamma _m(t)<L(t). \end{aligned}$$

As in the proof of Lemma 2.3 in Du et al. (2015), we can show that $x_i:=\lim _{t\nearrow t_1}\gamma _i(t)$ exists, and each $x_i$ is a zero of $w(t_1,\cdot )$, and the set $\{x_1,\ldots , x_m, L\}$ contains all the zeros of $w(t_1, \cdot )$ in $[l(t_1), L(t_1)]=[l(t_1), L]$.

We claim that $x_m=L$. Otherwise $x_m<L$ and $w(t,x)<0$ for $x\in (\gamma _m(t), L(t))$, $t\in [{\tilde{t}}_1, t_1]$. Since $w(t_1, L(t_1))=w(t_1, L)=0$, by the Hopf boundary lemma we deduce $w_x(t_1, L)>0$. It follows that

$$\begin{aligned} h'(t_1)=-\mu u_x(t_1, L)+\alpha a_-(L)=-\mu w_x(t_1, L)<0, \end{aligned}$$

a contradiction to (5.11). Hence $x_m=L$.

It is possible that several elements in the set $\{x_1,\ldots , x_m\}$ is identical to L. Let i be the smallest index such that $x_i=L$. Then $1\le i\le m$ and $w(t_1, x)$ has a fixed sign for $x\in [x_i-\delta , x_i)=[L-\delta , L)$, where $\delta >0$ is sufficiently small.

If $w(t_1, x)>0$ for $x\in [L-\delta , L)$, then by continuity we can find $\epsilon _1>0$ small so that $w(t, L-\delta )>0$ for $t\in [t_1, t_1+\epsilon _1]$. Then

$$\begin{aligned} (\underline{u}(t,x), \underline{h}(t)):=(V(x), L) \end{aligned}$$

is a lower solution to the problem satisfied by (u, h) over the region $x\in [L-\delta , L]$, $t\in [t_1, t_1+\epsilon _1]$. It follows that $h(t)\ge L$ and $u(t,x)\ge V(x)$ in this region, and by the strong maximum principle we further deduce $h(t)>L$, $u(t,x)>V(x)$ for $t\in (t_1, t_1+\epsilon _1]$, $x\in [L-\delta , L]$. Thus, applying Lemma 5.4 for $x\in [l(t), L-\delta ]$ we obtain, for $t\in (t_1, t_1+\epsilon _1]$,

$$\begin{aligned} \begin{aligned} Z[l(t), L(t)]&=Z[l(t), L]=Z[l(t), L-\delta ]+Z[L-\delta , L]=Z[l(t),L-\delta ]\\&\le Z[l(t_1), L-\delta ]=Z[l(t_1), L]-1\le m-1. \end{aligned} \end{aligned}$$

If $w(t_1, x)<0$ for $x\in [L-\delta , L)$, then by continuity we can find $\epsilon _1>0$ small so that $w(t, L-\delta )<0$ for $t\in [t_1, t_1+\epsilon _1]$. Then

$$\begin{aligned} (\overline{u}(t,x), \overline{h}(t)):=(V(x), L) \end{aligned}$$

is an upper solution to the problem satisfied by (u, h) over the region $x\in [L-\delta , h(t)]$, $t\in [t_1, t_1+\epsilon _1]$. It follows that $h(t)\le L$ and $u(t,x)\le V(x)$ in this region, and by the strong maximum principle we further deduce $h(t)<L$, $u(t,x)<V(x)$ for $t\in (t_1, t_1+\epsilon _1]$, $x\in [L-\delta , h(t)]$. Thus we again obtain

$$\begin{aligned} Z[l(t), L(t)]\le m-1 \quad \text{ for } \quad t\in (t_1, t_1+\epsilon _1]. \end{aligned}$$

(5.12)

By our assumption, there exists $t_2>t_1+\epsilon _1$ such that $h(t)-L\not =0$ for $t\in (t_1, t_2)$ and $h(t_2)-L=0$. We thus have

$$\begin{aligned} Z[l(t), L(t)]\le Z[l(t_1+\epsilon _1), L(t_1+\epsilon _1)]\le m-1 \quad \text{ for } \quad t\in (t_1+\epsilon _1, t_2). \end{aligned}$$

By an analogous analysis to the one leading to (5.12), we can find $\epsilon _2>0$ small such that

$$\begin{aligned} Z[l(t), L(t)]\le Z[l(t_1+\epsilon _1), L(t_1+\epsilon _1)]-1\le m-2 \quad \text{ for } \quad t\in (t_2, t_2+\epsilon _2]. \end{aligned}$$

(Modifications are needed if $h(t)>L$ for $t\in (t_1, t_2)$, but they are rather obvious.)

Since Z[l(t), L(t)] is nonnegative and m is a finite integer, the above process can be continued at most m steps, and at the last step, say step k, we conclude that $w(t,\cdot )$ does not change sign over [l(t), L(t)] for $t\in (t_k, t_k+\epsilon _k]$. Thus necessarily $w(t,x)<0$ for $x\in [l(t), L(t)]$ and $t\in (t_k, t_k+\epsilon _k]$. We may then compare (u(t, x), h(t)) with (V(x), L) over $x\in [l(t), h(t)]$, $t\ge t_k+\epsilon _k$ and apply the strong maximum principle, as before, to conclude that $u(t,x)<V(x),\; h(t)<L$ for $t\ge t_k+\epsilon _k$ and $x\in [l(t), h(t)]$, which is in contradiction to the assumption that $h(t)-L$ changes sign infinitely many times as $t\rightarrow \infty$. This completes the proof of Lemma 3.3 for the case that V(x) falls into case (i). $\square$

1.3.2 Proof of Lemma 3.3 when V(x) falls into case (ii)

The proof for this case follows the arguments above, except that we have to define l(t) differently, which leads to a case that requires attention. We now define

$$\begin{aligned} l(t):=l\in [0, L). \end{aligned}$$

If $h(t)>l$ for all $t>0$, then $w(t, l(t))=u(t, l)>0$ for all $t>0$, and we may follow the argument in the proof of case (i) above to conclude, with obvious minor changes.

Consider next the remaining case that $h(t_0)<l$ for some $t_0>0$. If $h(t)<L$ for all $t>t_0$, then the conclusion of the lemma already holds. Otherwise, there exist $t_2>t_1>t_0$ such that

$$\begin{aligned} h(t_1)=l,\; h(t_2)=L,\; l<h(t)<L \;\forall t\in (t_1, t_2). \end{aligned}$$

Since $V'(l)>0=V(l)$ and $u_x(t, h(t))<0=u(t, h(t))$, for $t>t_1$ but close to $t_1$, $w(t,\cdot )$ has a unique nondegenerate zero in (l, h(t)). Since

$$\begin{aligned} w(t,l)=u(t,l)>0,\; w(t, h(t))=-V(h(t))<0 \text{ for } t\in (t_1, t_2), \end{aligned}$$

we can apply Lemma 5.4 to conclude that

$$\begin{aligned} Z[l, h(t)]=1 \text{ for } t\in (t_1, t_2), \end{aligned}$$

and the unique zero of $w(t,\cdot )$ in [l, h(t)] is nondegenerate. Thus if we denote it by $\gamma (t)$, then it is a $C^1$ function of t in $(t_1, t_2)$, and $l<\gamma (t)<h(t)$. As before we know from Du et al. (2015) that $x_0:=\lim _{t\nearrow t_2}\gamma (t)$ exists and is a zero of $w(t_2,\cdot )$.

We claim that $x_0=L=h(t_2)$. Otherwise $x_0<L$, and by applying the maximum principle to the equation satisfied by w over $\{(t,x): \gamma (t)<x<h(t), \; t\in (t_1, t_2]\}$, we deduce that $w(t,x)<0$ over this region and $w_x(t_2, L)>0=w(t_2, L)$. It follows that

$$\begin{aligned} h'(t_2)=-\mu u_x(t_2, h(t_2))+\alpha a_-(L)=-\mu w_x(t_2, L)<0. \end{aligned}$$

On the other hand, from $h(t)<L$ for $t\in (t_1, t_2)$ we deduce $h'(t_2)\ge 0$. This controdiction proves our claim that $x_0=L$. We may now use the maximum principle to the equation of w over $\{(t,x): l<x<\gamma (t), t_1<t\le t_2\}$ to deduce that $w(t,x)>0$ in this region. It follows that $(\underline{u}(t,x), \underline{h}(t)):=(V(x), L)$ is a lower solution to the problem for (u, h) over the region $\{(t,x): l<x<L, t\ge t_2\}$, which implies, by the strong maximum principle, $h(t)>L$ and $u(t,x)>V(x)$ in this region. Hence $h(t)-L$ is positive for all large t, and the conclusion of the lemma holds. $\square$

1.3.3 Proof of Lemma 3.3 when V(x) falls into case (iii)

In this case, we divide the proof into three subcases:

$$\begin{aligned} \mathrm{(iii0):} \; V'(0)=0,\;\; \mathrm{(iii+):} \; V'(0)>0,\;\; \mathrm{(iii-):} \; V'(0)<0. \end{aligned}$$

In case (iii0), V(x) is an even function of x and so it w(t, x). The arguments used for case (i) carry over easily if we define $l(t)=-L(t)$.

In the remaining two cases, some extra analysis is needed. We first consider case (iii+). We again want to follow the analysis of case (i) but this time we have to define l(t) rather differently, according to the behavior of w(t, 0) as t increases to infinity.

If $w(t,0)\not =0$ for all large t, then we take $l(t)=0$ and the proof in case (i) can be used without change. So we only need to consider the case that

$$\begin{aligned} w(t,0)=0 \,\text{ for } \text{ infinitely } \text{ many } \text{ large }\, t. \end{aligned}$$

We observe that $w_x(t,0)=-V'(0)<0$ for all $t>0$. Moreover, by standard parabolic and elliptic estimates for u(t, x) and V(x), respectively, there exists $C>0$ independent of $t\ge 1$ such that

$$\begin{aligned} |w_{xx}(t,x)|,\; |w_t(t,x)|\le C \text{ for } t\ge 1,\; |x|\le \min \{h_0, h_a, -l, L\}. \end{aligned}$$

(5.13)

Hence there exists $\epsilon _0>0$ small such that

$$\begin{aligned} w_x(t,x)\le -V'(0)/2 \hbox { for } t\ge 1\hbox { and }|x|\le \epsilon _0. \end{aligned}$$

Define

$$\begin{aligned} J:=\{t>1: w(t,0)>0\}. \end{aligned}$$

If there exists $t_0> 1$ such that $J\cap [t_0,\infty )=\emptyset$, then $w(t,0)\le 0$ for $t>t_0$ and hence $w(t,\epsilon _0)<0$ for all $t>t_0$. We may then take $l(t)=\epsilon _0$ and the proof of case (i) can be used without change.

So we assume that $J\cap [t_0,\infty )\not =\emptyset$ for all $t_0>1$. Since we have assumed that $w(t,0)=0$ along a sequence of t going to $\infty$, the open set J is necessarily the union of infinitely many disjoint open intervals:

$$\begin{aligned} J=\cup _{i\in I}J_i,\; J_i=(s_i, t_i),\; \text{ where } \text{ the } \text{ index } \text{ set }\, I \text{ contains } \text{ infinitely } \text{ many } \text{ elements. } \end{aligned}$$

Set

$$\begin{aligned} \sigma _0:=\frac{V'(0)\epsilon _0}{4C}. \end{aligned}$$

If $|t_i-s_i|\le \sigma _0$, then, due to $w(s_i,0)=w(t_i,0)=0$, we have

$$\begin{aligned} w(t, 0)\le \frac{\sigma _0 C}{2} \text{ for } t\in (s_i, t_i), \end{aligned}$$

and hence

$$\begin{aligned} w(t, \epsilon _0/2)\le \frac{\sigma _0 C}{2}-\frac{V'(0)\epsilon _0}{4}<0 \text{ for } t\in [s_i, t_i]. \end{aligned}$$

Define

$$\begin{aligned} I^0:=\{i\in I: |t_i-s_i|>\sigma _0\}. \end{aligned}$$

If $I^0$ is a finite set, then for all large t, $w(t, \epsilon _0/2)<0$, and hence we can take $l(t)=\epsilon _0/2$ and argue as in case (i) to finish the proof.

Suppose from now on that $I^0$ is an infinite set. Then the set of intervals $\{J_i\}_{i\in I^0}$ can be ordered, namely, we can take $I^0=\{1,2,\ldots \}$ and require $t_i<s_{i+1}$ for $i=1,2,\ldots$ Set

$$\begin{aligned} J^0:=\cup _{i\in I^0}(s_i, t_i). \end{aligned}$$

Then for any $t\in (1,\infty )\backslash J^0$, we have $w(t, \epsilon _0/2)<0$. Using the ordering property of the intervals in $J^0$, we now see that

$$\begin{aligned} w(t, \epsilon _0/2)<0 \text{ for } t\in [t_i, s_{i+1}],\; w(t, 0)>0 \text{ for } t\in (s_i, t_i),\; i=1,2,\ldots \end{aligned}$$

By the monotonisity of w(t, x) in x for $|x|\le \epsilon _0$, we find that

$$\begin{aligned} w(t, -\epsilon _0/2)>0 \text{ for } t\in [s_i, t_i]. \end{aligned}$$

We now examine the zero number of w(t, x) for $x\in [-\epsilon _0/2, L(t)]$ when t is in a small neighborhood of $[s_i, t_i]$, which we denote by $Z_i[-\epsilon _0/2, L(t)]$, and the zero number of w(t, x) for $x\in [\epsilon _0/2, L(t)]$ when t belongs to a small neighborhood of $[t_i, s_{i+1}]$, which we denote by $Z_i[\epsilon _0/2, L(t)]$.

We note that $w(t, -\epsilon _0/2)>0$ for t in a small neighborhood of $[s_i, t_i]$ and $w(t,\epsilon _0/2)<0$ for t in a small neighborhood of $[t_i, s_{i+1}]$. Therefore, for all large i, by our arguments in the proof of case (i), $Z_i[-\epsilon _0/2, L(t)]$ is finite for t in a small neighborhood of $[s_i, t_i]$, and each time $h(t)-L$ changes sign, say at ${\tilde{t}}\in [s_i, t_i]$, the value of $Z_i[-\epsilon _0/2, L(t)]$ is decreased by at least 1 when t increases across ${\tilde{t}}$. The same also holds for $Z_i[\epsilon _0/2, L(t)]$ for t in a small neighborhood of $[t_i, s_{i+1}]$.

For t close to $t_i$, clearly

$$\begin{aligned} Z_i[-\epsilon _0/2, L(t)]=1+Z_i[\epsilon _0/2, L(t)]. \end{aligned}$$

For t close to $s_{i+1}$, we have

$$\begin{aligned} Z_i[\epsilon _0/2, L(t)]=Z_{i+1}[-\epsilon _0/2, L(t)]-1. \end{aligned}$$

Suppose by way of contradiction that $h(t)-L$ changes sign infinitely many times as $t\rightarrow \infty$, say at $t_n$ with $t_n$ increasing to infinity. Then by the zero number consideration, no one interval of the form $[s_i, t_i]$, or of the form $[t_i, s_{i+1}]$, can contain infinitely many members from $\{t_n\}$. Thus there are infinitely many intervals of the form $[s_i, t_i]$ such that each of them contains at least one member of $\{t_n\}$, or infinitely many intervals of the form $[t_i, s_{i+1}]$ such that each contains at least one member of $\{t_n\}$. For definiteness, we assume that there exist $i_k\rightarrow \infty$ and $n_k\rightarrow \infty$ such that $t_{n_k}\in [s_{i_k}, t_{i_k}]$, $i_{k}<i_{k+1}$, $n_k<n_{k+1}$, $k=1,2,\ldots$ Then by our analysis above, we have

$$\begin{aligned} Z_{i_k}[-\epsilon _0/2, L(t_{i_k})]\le Z_{i_1}[-\epsilon _0/2, L(t_{i_1})]-k+1<0 \end{aligned}$$

for all large k. This contradiction shows that $h(t)-L$ does not change sign for all large t. This completes our proof of Lemma 3.3 for the case that V(x) falls in case (iii+).

The proof for case (iii−) is analogous, where we replace the definition of J by

$$\begin{aligned} J:=\{t>1: w(t,0)<0\}, \end{aligned}$$

and then make obvious changes. We omit the details. $\square$