Abstract
We investigate the spreading behavior of two invasive species modeled by a Lotka–Volterra diffusive competition system with two free boundaries in a spherically symmetric setting. We show that, for the weak–strong competition case, under suitable assumptions, both species in the system can successfully spread into the available environment, but their spreading speeds are different, and their population masses tend to segregate, with the slower spreading competitor having its population concentrating on an expanding ball, say \(B_t\), and the faster spreading competitor concentrating on a spherical shell outside \(B_t\) that disappears to infinity as time goes to infinity.
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1 Introduction
In this paper, we investigate the spreading behavior of two competing species described by the following free boundary problem in \(\mathbb {R}^N\) (\(N\ge 1\)) with spherical symmetry:
where u(r, t) and v(r, t) represent the population densities of the two competing species at spatial location \(r\, (=|x|)\) and time t; \(\Delta \varphi :=\varphi _{rr}+\frac{(N-1)}{r}\phi _r\) is the usual Laplace operator acting on spherically symmetric functions. All the parameters are assumed to be positive, and without loss of generality, we have used a simplified version of the Lotka–Volterra competition model, which can be obtained from the general model by a standard change of variables procedure (see, for example, [13]). The initial data \((u_0,v_0,s_1^0,s_2^0)\) satisfies
In this model, both species invade the environment through their own free boundaries: the species u has a spreading front at \(r=s_1(t)\), while v’s spreading front is at \(r=s_2(t)\). For the mathematical treatment, we have extended u(r, t) from its population range \(r\in [0, s_1(t)]\) to \(r>s_1(t)\) by 0, and extended v(r, t) from \(r\in [0, s_2(t)]\) to \(r>s_2(t)\) by 0.
The global existence and uniqueness of the solution to problem (P) under (1.1) can be established by the approach in [15] with suitable changes. In fact, the local existence and uniqueness proof can cover a rather general class of such free boundary systems. The assumption in (P) that u and v have independent free boundaries causes some difficulties but this can be handled by following the approach in [15] with suitable modifications and corrections. The details are given in the Appendix at the end of the paper.
We say \((u,v,s_1, s_2)\) is a (global classical) solution of (P) if
where
and all the equations in (P) are satisfied pointwisely. By the Hopf boundary lemma, it is easily seen that, for \(i=1,2\) and \(t>0\), \(s_i'(t)>0\). Hence
is well-defined.
We are interested in the long-time behavior of (P). In order to gain a good understanding, we focus on some interesting special cases. Our first assumption is that
It is well known that under this assumption, when restricted over a fixed bounded domain \(\Omega \) with no-flux boundary conditions, the unique solution \((\tilde{u}(x,t), \tilde{v}(x,t))\) of the corresponding problem of (P) converges to (1, 0) as \(t\rightarrow \infty \) uniformly for \(x\in \overline{\Omega }\). So in the long run, the species u drives v to extinction and wins the competition. For this reason, condition (1.2) is often referred to as the case that u is superior and v is inferior in the competition. This is often referred to as a weak–strong competition case. A symmetric situation is \(0<h<1<k\).
The case \(h,k\in (0,1)\) is called the weak competition case (see [33]), while the case \(h,k\in (1,+\infty )\) is known as the strong competition case. In these cases, rather different long-time dynamical behaviors are expected.
In this paper, we will focus on problem (P) for the weak–strong competition case (1.2), and demonstrate a rather interesting phenomenon, where u and v both survive in the competition, but they spread into the new territory with different speeds, and their population masses tend to segregate, with the population mass of v shifting to infinity as \(t\rightarrow \infty \).
For (P) with space dimension \(N=1\), such a phenomenon was discussed in [15], though less precisely than here. It is shown in Theorem 5 of [15] that under (1.2) and some additional conditions, both species can spread successfully, in the sense that
-
(i)
\(s_{1,\infty }=s_{2,\infty }=\infty \),
-
(ii)
there exists \(\delta >0\) such that for all \(t>0\),
$$\begin{aligned} u(x,t)\ge \delta \hbox { for }x\in I_u(t),\; v(x,t)\ge \delta \hbox { for }x\in I_v(t), \end{aligned}$$where \(I_u(t)\) and \(I_v(t)\) are intervals of length at least \(\delta \) that vary continuously in t.
At the end of the paper [15], the question of determining the spreading speeds for both species was raised as an open problem for future investigation.
In this paper, we will determine, for such a case, \( \lim _{t\rightarrow \infty }\frac{s_i(t)}{t},\; i=1,2\); so in particular, the open problem of [15] on the spreading speeds is resolved here. Moreover, we also obtain a much better understanding of the long-time behavior of \(u(\cdot , t)\) and \(v(\cdot , t)\), for all dimensions \(N\ge 1\). See Theorem 1 and Corollary 1 below for details.
A crucial new ingredient (namely \(c^*_{\mu _1}\) below) in our approach here comes from recent research on another closely related problem (proposed in [7]):
where \(0<k<1<h\) and
In problem (Q) the inferior competitor v is assumed to be a native species already established in the environment, while the superior competitor u is invading the environment via the free boundary \(r=h(t)\). Theorem 4.3 in [7] gives a spreading-vanishing dichotomy for (Q): Either
-
(Spreading of u) \(\lim _{t\rightarrow \infty } h(t)=\infty \) and
$$\begin{aligned} \lim _{t\rightarrow \infty } (u(r, t), v(r,t))=(1,0) \text{ locally } \text{ uniformly } \text{ for } \,r\in [0,\infty ), \hbox { or} \end{aligned}$$ -
(Vanishing of u) \(\lim _{t\rightarrow \infty } h(t)<\infty \) and
$$\begin{aligned} \lim _{t\rightarrow \infty } (u(r, t), v(r,t))=(0,1) \text{ locally } \text{ uniformly } \text{ for } \,r\in [0,\infty ). \end{aligned}$$
Sharp criteria for spreading and vanishing of u are also given in [7]. When spreading of u happens, an interesting question is whether there exists an asymptotic spreading speed, namely whether \(\lim _{t\rightarrow \infty }\frac{h(t)}{t}\) exists. This kind of questions, similar to the one being asked in [15] mentioned above, turns out to be rather difficult to answer for systems of equations with free boundaries. Recently, Du, Wang and Zhou [13] successfully established the spreading speed for (Q), by making use of the following so-called semi-wave system:
It was shown that (1.4) has a unique solution if \(c\in [0, c_0)\), and it has no solution if \(c\ge c_0\), where
is the minimal speed for the traveling wave solution studied in [20]. More precisely, the following result holds:
Theorem A
(Theorem 1.3 of [13]) Assume that \(0<k<1<h\). Then for each \(c\in [0,c_0)\), (1.4) has a unique solution \((U_c,V_c)\in [C(\mathbb {R})\cap C^2([0,\infty ))]\times C^2(\mathbb {R})\), and it has no solution for \(c\ge c_0\). Moreover,
-
(i)
if \(0\le c_1< c_2<c_0\), then
$$\begin{aligned} U'_{c_1}(0)<U'_{c_2}(0),\; U_{c_1}(\xi )>U_{c_2}(\xi ) \text{ for } \xi <0,\; V_{c_2}(\xi )>V_{c_1}(\xi ) \text{ for } \xi \in \mathbb {R}; \end{aligned}$$ -
(ii)
the mapping \(c\mapsto (U_c,V_c)\) is continuous from \([0,c_0)\) to \(C^2_{loc}((-\infty ,0])\times C^2_{loc}(\mathbb {R})\) with
$$\begin{aligned} \lim _{c\rightarrow c_0}(U_c,V_c)=(0,1)\quad \text{ in }\, C^2_{loc}((-\infty ,0])\times C^2_{loc}(\mathbb {R}); \end{aligned}$$ -
(iii)
for each \(\mu _1>0\), there exists a unique \(c=c^*_{\mu _1}\in (0,c_0)\) such that
$$\begin{aligned} \mu _1 U'_{c^*_{\mu _1}}(0)=c^*_{\mu _1}\quad \text{ and } c^*_{\mu _1}\nearrow c_0\hbox { as }{\mu _1}\nearrow \infty . \end{aligned}$$
The spreading speed for (Q) is established as follows.
Theorem B
(Theorem 1.1 of [13]) Assume that \(0<k<1<h\). Let (u, v, h) be the solution of (Q) with (1.3) and
If \(h_{\infty }:=\lim _{t\rightarrow \infty }h(t)=\infty \), then
where \(c^*_{\mu _1}\) is given in Theorem A.
It turns out that \(c^*_{\mu _1}\) also plays an important role in determining the long-time dynamics of (P). In order to describe the second crucial number for the dynamics of (P) (namely \(s^*_{\mu _2}\) below), let us recall that, in the absence of the species u, problem (P) reduces to a single species model studied by Du and Guo [2], who generalized the model proposed by Du and Lin [6] from one dimensional space to high dimensional space with spherical symmetry. In such a case, a spreading-vanishing dichotomy holds for v, and when spreading happens, the spreading speed of v is related to the following problem
More precisely, by Proposition 2.1 in [1] (see also Proposition 1.8 and Theorem 6.2 of [8]), the following result holds:
Theorem C
For fixed \(a,b,d,\mu _2>0\), there exists a unique \(s=s^*(a,b,d,\mu _2)\in (0, 2\sqrt{ad})\) and a unique solution \(q^*\) to (1.6) with \(s=s^*(a,b,d,\mu _2)\) such that \((q^*)'(0)=-s^*(a,b,d,\mu _2)/\mu _2\). Moreover, \((q^*)'(\xi )<0\) for all \(\xi \le 0\).
Hereafter, we shall denote \(s^*_{\mu _2}:=s^*(1,1,1,\mu _2)\). It turns out that the long-time behavior of (P) depends crucially on whether \(c^*_{\mu _1}<s^*_{\mu _2}\) or \(c^*_{\mu _1}>s^*_{\mu _2}\). As demonstrated in Theorems 1 and 2 below, in the former case, it is possible for both species to spread successfully, while in the latter case, at least one species has to vanish eventually.
Let us note that while the existence and uniqueness of \(s^*_{\mu _2}\) is relatively easy to establish (and has been used in [15] and other papers to estimate the spreading speeds for various systems), this is not the case for \(c^*_{\mu _1}\), which takes more than half of the length of [13] to establish. The main advance of this research from [15] is achieved by making use of \(c^*_{\mu _1}\).
Theorem 1
Suppose (1.2) holds and
Then one can choose initial functions \(u_0\) and \(v_0\) properly such that the unique solution \((u,v, s_1, s_2)\) of (P) satisfies
and for every small \(\epsilon >0\),
Before giving some explanations regarding the condition (1.7) and the choices of \(u_0\) and \(v_0\) in the above theorem, let us first note that the above conclusions indicate that the u species spread at the asymptotic speed \(c^*_{\mu _1}\), while v spreads at the faster asymptotic speed \(s^*_{\mu _2}\). Moreover, (1.8) and (1.9) imply that the population mass of u roughly concentrates on the expanding ball \(\{r<c^*_{\mu _1}t\}\), while that of v concentrates on the expanding spherical shell \(\{c^*_{\mu _1}t<r< s^*_{\mu _2}t\}\) which shifts to infinity as \(t\rightarrow \infty \). We also note that, apart from a relatively thin coexistence shell around \(r=c^*_{\mu _1}t\), the population masses of u and v are largely segregated for all large time. Clearly this gives a more precise description for the spreadings of u and v than that in Theorem 5 of [15] (for \(N=1\)) mentioned above.
We now look at some simple sufficient conditions for (1.7). We note that \(c^*_{\mu _1}\) is independent of \(\mu _2\) and the initial functions. From the proof of Lemma 2.9 in [13], we see that \(c^*_{\mu _1}\rightarrow 0\) as \(\mu _1\rightarrow 0\). Therefore when all the other parameters are fixed,
A second sufficient condition can be found by using Theorem A (iii), which implies \(c^*_{\mu _1}< c_0\le 2\sqrt{rd}\) for all \(\mu _1>0\). It follows that
Note that \(2\sqrt{rd}\le s^*_{\mu _2}\) holds if \(\sqrt{rd}<1\) and \(\mu _2\gg 1\) since \(s^*_{\mu _2}\rightarrow 2\) as \(\mu _2\rightarrow \infty \).
For the conditions in Theorem 1 on the initial functions \(u_0\) and \(v_0\), the simplest ones are given in the corollary below.
Corollary 1
Assume (1.2) and (1.7). Then there exists a large positive constant \(C_0\) depending on \(s^0_1\) such that the conclusions of Theorem 1 hold if
-
(i)
\(\Vert u_0\Vert _{L^{\infty }([0,s^0_1])}\le 1\) with \(s^0_1\ge R^*\sqrt{d/[r(1-k)]}\),
-
(ii)
for some \(x_0\ge C_0\) and \( L\ge C_0\), \(v_0(r)\ge 1\) for \(r\in [x_0,x_0+L]\).
Here \(R^*\) is uniquely determined by \(\lambda _1(R^*)=1\), where \(\lambda _1(R)\) is the principal eigenvalue of
Roughly speaking, conditions (i) and (ii) above (together with (1.2) and (1.7)) guarantee that u does not vanish yet it cannot spread too fast initially, and the initial population of v is relatively well-established in some part of the environment where u is absent, so with its fast spreading speed v can outrun the superior but slower competitor u. In Sect. 2, weaker sufficient conditions on \(u_0\) and \(v_0\) will be given (see (B1) and (B2) there).
Next we describe the long-time behavior of (P) for the case
We will show that, in this case, no matter how the initial functions \(u_0\) and \(v_0\) are chosen, at least one of u and v will vanish eventually. As in [15], we say u (respectively v) vanishes eventually if
and we say u (respectively v) spreads successfully if
where \(I_u(t)\) is an interval of length at least \(\delta \) that varies continuously in t (respectively,
where \(I_v(t)\) is an interval of length at least \(\delta \) that varies continuously in t).
For fixed \(d,r,h,k>0\) satisfying (1.2), we define
Note that \(\mathcal {B}\ne \emptyset \) since \(s^*_{\mu _2}\rightarrow 0\) as \(\mu _2\rightarrow 0\) and \(c^*_{\mu _1}>0\) is independent of \(\mu _2\).
We have the following result.
Theorem 2
Assume that (1.2) holds. If \((\mu _1,\mu _2)\in \mathcal {B}\), then at least one of the species u and v vanishes eventually. More precisely, depending on the choice of \(u_0\) and \(v_0\), exactly one of the following occurs for the unique solution \((u,v,s_1,s_2)\) of (P):
-
(i)
Both species u and v vanish eventually.
-
(ii)
The species u vanishes eventually and v spreads successfully.
-
(iii)
The species u spreads successfully and v vanishes eventually.
Note that \((\mu _1,\mu _2)\in \mathcal {B}\) if and only if (1.10) holds. Theorem 2 can be proved along the lines of the proof of [15, Corollary 1] with some suitable changes. When \(N=1\), Theorem 2 slightly improves the conclusion of Corollary 1 in [15], since it is easily seen that \(\mathcal {A}\subset \mathcal {B}\) (due to \(s^*(r(1-k),r,d,\mu _1)\le c^*_{\mu _1}\)), where \(\mathcal {A}:=\Big \{(\mu _1,\mu _2)\in \mathbb {R}_+\times \mathbb {R}_+: \ s^*(r(1-k),r,d,\mu _1)>s^*_{\mu _2}\Big \}\) is given in [15].
Remark 1.1
We note that by suitably choosing the initial functions \(u_0\) and \(v_0\) and the parameters \(\mu _1\) and \(\mu _2\), all the three possibilities in Theorem 2 can occur. For example, for given \(u_0\) and \(v_0\) with \(s_1^0<R^*\sqrt{\frac{d}{r}}\) and \(s_2^0< R^*\), then scenario (i) occurs as long as both \(\mu _1\) and \(\mu _2\) are small enough and \((\mu _1, \mu _2)\in \mathcal {B}\) (which can be proved by using the argument in [6, Lemma 3.8]). If next we modify \(v_0\) such that \(s_2^0\ge R^*\), then u still vanishes eventually but v will spread successfully, which leads to scenario (ii). For scenario (iii) to occur, we can take \(s_1^0\ge R^*\sqrt{\frac{d}{r(1-k)}}\) and \(\mu _2\) small enough.
Our results here suggest that in the weak–strong competition case, co-existence of the two species over a common (either moving or stationary) spatial region can hardly happen. This contrasts sharply to the weak competition case (\(h,k\in (0,1)\)), where coexistence often occurs; see, for example [31, 33].
Before ending this section, we mention some further references that form part of the background of this research. Since the work [6], there have been tremendous efforts towards developing analytical tools to deal with more general single species models with free boundaries; see [1, 3,4,5, 8, 9, 11, 12, 17, 19, 21, 22, 24, 25, 27, 35] and references therein. Related works for two species models can be found in, for example, [7, 13, 14, 26, 28,29,30, 32,33,34]. The issue of the spreading speed for single species models in homogeneous environment has been well studied, and we refer to [11, 12] for some sharp estimates. Some of the theory on single species models can be used to estimate the spreading speed for two species models; however, generally speaking, only rough upper and lower bounds can be obtained via this approach.
The rest of this paper is organized as follows. In Sect. 2, we shall prove our main result, Theorem 1, based on the comparison principle and on the construct of various auxiliary functions as comparison solutions to (P). Section 3 is an appendix, where we prove the local and global existence and uniqueness of solutions to a wide class of problems including (P) as a special case, and we also sketch the proof of Theorem 2.
2 Proof of Theorem 1
We start by establishing several technical lemmas.
Lemma 2.1
Let \(\mu _2>0\) and \(s^*_{\mu _2}\) be given in Theorem C. Then for each \(s\in (0,s^*_{\mu _2})\), there exists a unique \(z=z(s)>0\) such that the solution \(q_s\) of the initial value problem
satisfies \(q_s'(-z(s))=0\) and \(q_s'(z)<0\) for \(z\in (-z(s),0)\). Moreover, \(q_s(-z(s))\) is continuous in s and
Proof
The conclusions follow directly from Proposition 2.4 in [18]. \(\square \)
Lemma 2.2
Let \((u,v,s_1,s_2)\) be a solution of (P) with \(s_{1,\infty }=s_{2,\infty }=\infty \). Suppose that
for some positive constants \(c_1\) and \(c_2\). Then for any \(\varepsilon >0\), there exists \(T>0\) such that
Proof
Let \(\bar{w}\) be the solution of \(w'(t)=w(1-w)\) with initial data \(w(0)=\Vert v_0\Vert _{L^{\infty }}.\) By the standard comparison principle, \(v(x,t)\le \bar{w}(t)\) for all \(t\ge 0\). Since \(\bar{w}\rightarrow 1\) as \(t\rightarrow \infty \), there exists \(T>0\) such that (2.1) holds.
Before proving (2.2), we first show \(\limsup _{t\rightarrow \infty }s_2(t)/t\le s^*_{\mu _2}\) by simple comparison. Indeed, it is easy to check that \((v,s_2)\) forms a subsolution of
By the comparison principle (Lemma 2.6 of [2]), \(\bar{\eta }(t)\ge s_{2}(t)\) for all t, which implies that \(\bar{\eta }(\infty )=\infty \). It then follows from Corollary 3.7 of [2] that \(\bar{\eta }(t)/t\rightarrow s^*_{\mu _2}\) as \(t\rightarrow \infty \). Consequently, we have
It follows that \(c_2<s^{*}_{\mu _2}\).
We now prove (2.2) by using a contradiction argument. Assume that the conclusion does not hold. Then there exist small \(\epsilon _0>0\), \(t_k \uparrow \infty \) and \(x_k \in [c_1 t_k, c_2 t_k]\) such that
Up to passing to a subsequence we may assume that \(p_k:=x_k/t_k \rightarrow p_0\) for some \(p_0 \in [c_1,c_2]\) as \(k\rightarrow \infty \).
We want to show that
which would give the desired contradiction (with (2.3)). To do so, we define
Then \(w_k\) satisfies
Recall that \(0<c_1<c_2<s_{\mu _2}^*<2\), \(x_k=p_kt_k\) and \(p_k\rightarrow p_0\in [c_1, c_2]\subset (0,2)\). Hence there exists large positive L such that for all \(L_1\), \(L_2\in [L, \infty )\), the problem
has a unique positive solution z(R) and \(z(0)>1-\epsilon _0\).
Fix \(L_1\ge L\), \(p\in (p_0, 2)\) and define
It is easily checked that
Moreover, there exists a unique \(R_0\in (-L_1, 0)\) such that
We may assume that \(L_1\) is large enough such that
We then choose \(L_2>L\) such that
Set
Then clearly
Since
for all large k and large t, we further obtain, for such k and t, say \(k\ge k_0\) and \(t\ge T_1\),
The above differential inequality should be understood in the weak sense since \(\tilde{\phi }''\) may have a jump at \(R=R_0\).
We now fix \(T_0\ge T_1\) and observe that
Hence if \(T_0\) is large enough then for \(R\in [-L_2, L_1]\) and \(t\ge T_0\) we have
It follows that
Let \(z_k(R,t)\) be the unique solution of
with initial condition
The comparison principle yields
since \(w_k(R, t)>0=z_k(R,t)\) for \(R\in \{-L_2, L_1\}\) and \(t> T_0,\; k\ge 1\).
On the other hand, if we choose \(\delta >0\) sufficiently small, then \(\underline{z}(R):=\delta \tilde{\phi }(R)\le \sigma _0\) for \(R\in [-L_2, L_1]\) and due to (2.6), \(\underline{z}(R)\) satisfies
We thus obtain
We claim that
where z(R) is the unique positive solution of (2.5), which then gives
and so (2.4) holds.
It remains to prove (2.7). Set
Then \(Z_k\) satisfies
and
By a simple comparison argument involving a suitable ODE problem we easily obtain
Since \( \frac{N-1}{R+p_k(t_k+t)}+p_k\rightarrow p_0\) uniformly as \(k\rightarrow \infty \), we may apply the parabolic \(L^p\) estimate to the equations satisfied by \(Z_k\) to conclude that, for any \(p>1\) and \(T>0\), there exists \(C_1>0\) such that, for all large \(k\ge k_0\), say \(k\ge k_1\),
It then follows from the Sobolev embedding theorem that, for every \(\alpha \in (0,1)\) and all \(k\ge k_1\),
for some constant \(C_2\) depending on \(C_1\) and \(\alpha \). Let \(\tilde{\alpha }\in (0, \alpha )\). Then by compact embedding and a well known diagonal process, we can find a subsequence of \(\{Z_k\}\), still denoted by itself for the seek of convenience, such that
From the equations satisfied by \(Z_k\) we obtain
and
We show that \(Z(R,t)\equiv z(R)\). Indeed, if we denote by \(\underline{Z}\) the unique solution of
with boundary conditions \(z(-L_2,t)=z(L_1,t)=0\) and initial condition \(z(R,0)=\underline{z}(R)\), while let \(\overline{Z}\) be the unique solution to this problem but with initial condition replaced by \(z(R,0)=M\), then clearly
On the other hand, for any \(s>0\), by the comparison principle we have
Letting \(s\rightarrow \infty \) we obtain \(z(R)\le Z(R,t)\le z(R)\). We have thus proved \(Z(R,t)\equiv z(R)\) and hence
This proves (2.7) and the proof of Lemma 2.2 is complete.\(\square \)
We now start to construct some auxiliary functions by modifying the unique solution (U, V) of (1.4) with \(c=c_{\mu _1}^*\). Firstly, for any given small \(\varepsilon \in (0,1)\) we consider the following perturbed problem
Taking \(U=(1+\varepsilon )\widehat{U}\) and \(V=(1-\varepsilon )\widehat{V}\), then \((\widehat{U}, \widehat{V})\) satisfies (1.4) for k and h replaced by some \(\hat{k}_\varepsilon \) and \(\hat{h}_\varepsilon \) with \(0<\hat{k}_\varepsilon<1<\hat{h}_\varepsilon \), and \(\hat{k}_\varepsilon \rightarrow k\), \(\hat{h}_\varepsilon \rightarrow h\) as \(\varepsilon \rightarrow 0\). Hence by Theorem A, there exists a unique \(c=c^{\varepsilon }_{\mu _1}>0\) such that (2.8) with \(c=c^{\varepsilon }_{\mu _1}\) admits a unique solution \((U_{\varepsilon },V_{\varepsilon })\). As in [13], \((U_\varepsilon , V_\varepsilon )\) and \(c^\varepsilon _{\mu _1}\) depends continuously on \(\varepsilon \), and in particular, \(c^{\varepsilon }_{\mu _1}\rightarrow c^{*}_{\mu _1}\) as \(\varepsilon \rightarrow 0\). Moreover, as in the proof of Lemma 2.5 in [13], we have the asymptotic expansion
for some \(C>0\), where
Next we modify \((U_{\varepsilon }(\xi ),V_{\varepsilon }(\xi ))\) to obtain the required auxiliary functions. The modification of \(V_{\varepsilon }\) is rather involved, and for simplicity, we do that for \(\xi \ge 0\) and \(\xi \le 0\) separately.
We first consider the case \(\xi \ge 0\). For fixed \(\varepsilon \in (0,1)\) sufficiently small, we define
where \(\xi _0=\xi _0(\varepsilon )>0\) is determined later and
It is straightforward to see that \(Q^{\varepsilon }_+\in C^1([0,\xi _0+1])\). The following result will be useful later.
Lemma 2.3
For any small \(\varepsilon >0\), there exist \(\xi _0=\xi _0(\varepsilon )>0\) and \(\xi _1=\xi _1(\varepsilon )\in (\xi _0,\xi _0+1)\) such that \(\lim _{\varepsilon \rightarrow 0}\xi _0(\varepsilon )=\infty \) and
Moreover, there exists \(s^\varepsilon \in (0,s^*_{\mu _2})\) such that \(s^\varepsilon \rightarrow s^*_{\mu _2}\) as \(\varepsilon \rightarrow 0^+\) and
where \(z(s^\varepsilon )\) and \(q_{s^\varepsilon }\) are defined in Lemma 2.1 with \(s=s^\varepsilon \).
Proof
For convenience of notation we will write \(Q^{\varepsilon }_+=Q\). Since \(V_{\varepsilon }(\infty )= 1-\varepsilon \), \(V_{\varepsilon }'(\infty )=0\) and \(V_\varepsilon '>0\), we can choose \(\xi _0=\xi _0(\varepsilon )\gg 1\) such that \(\lim _{\varepsilon \rightarrow 0}\xi _0(\varepsilon )=\infty \) and
In particular, we have
Note that \(Q'(\xi _0)=V_{\varepsilon }'(\xi _0)>0\). By the continuity of \(Q'\), we can find \(\xi _1=\xi _1(\xi _0)\in (\xi _0,\xi _0+1)\) such that
Hence we have \(Q'>0\) in \([0,\xi _1)\) since \(Q'=V_{\varepsilon }'>0\) in \([0,\xi _0]\).
We now prove (2.12). For \(\xi \in [0,\xi _0)\), we have \(Q=V_{\varepsilon }\). Using \(U_{\varepsilon }(\xi )\equiv 0\) for \(\xi \ge 0\) and the second equation of (2.8), it is straightforward to see that the inequality in (2.12) holds for \(\xi \in [0,\xi _0)\). For \(\xi \in (\xi _0,\xi _1]\), direct computation gives us
Hence
Using \(0\le \xi -\xi _0\le \xi _1-\xi _0<1\), \(1>V_{\varepsilon }(\xi )>V_{\varepsilon }(\xi _0)\) for \(\xi \in [\xi _0,\xi _1]\), the identity
and (2.11), we deduce
To complete the proof, it remains to show the existence of \(s^{\varepsilon }\). Note that
By (2.14), we have
By Lemma 2.1, \(q_s(-z(s))\) is a continuous and increasing function of s for \(s\in (0, s^*_{\mu _2})\), and \(q_s(-z(s))\rightarrow 1\) as \(s\rightarrow s^*_{\mu _2}\). Therefore, in view of (2.15), for each small \(\varepsilon >0\) there exists \(s^\epsilon \in (0, s^*_{\mu _2})\) such that
Moreover, \(s^{\varepsilon }\rightarrow s^*_{\mu _2}\) as \(\varepsilon \rightarrow 0\). Thus (2.13) holds. The proof of Lemma 2.4 is now complete.\(\square \)
We now consider the case \(\xi \le 0\). We define
where
with \(\lambda >0\) and \(\xi _2<0\) to be determined below.
Lemma 2.4
Let \(\varepsilon >0\) be sufficiently small and \((U_\varepsilon ,V_\varepsilon )\) be the solution of (2.8) with \(c=c^{\varepsilon }_{\mu _1}\). Then there exist \(\lambda =\lambda (\varepsilon )>0\) sufficiently small and \(\xi _2=\xi _2(\varepsilon )<0\) such that \(V_{\varepsilon }(\xi _2)=Q^{\varepsilon }_{-}(\xi _2)<\varepsilon \) and
Moreover, there exists a unique \(\xi _3\in (-\infty ,\xi _2)\) depending on \(\xi _2\) and \(\lambda \) such that \(Q^{\varepsilon }_{-}(\xi _3)=0\) and the following inequality holds:
Proof
We write \(Q^{\varepsilon }_-=Q\) for convenience of notation. Using \(\gamma '(0)=0\), it is straightforward to see that
for any choice of \(\xi _2<0\). Since \(V'_{\varepsilon }>0\) in \(\mathbb {R}\) and \(\gamma '(\xi )>0\) for \(\xi <0\), we have
Hence (2.17) holds for any choice of \(\xi _2<0\).
For any given \(\lambda >0\), we take \(K_{\lambda }>0\) such that
where \(\mu >0\) is given in (2.9). By (2.9), we have
Together with (2.19), and \((U_{\varepsilon },V_{\varepsilon })(-\infty )=(1+\varepsilon ,0)\), we can take \(\xi _2=\xi _2(\lambda )\) close to \(-\infty \) such that
On the other hand, since \(Q(\xi _2)=V_{\varepsilon }(\xi _2)>0\), we can apply the intermediate value theorem to obtain \(\xi _3\in (\xi _2-K_{\lambda },\xi _2)\) such that \(Q(\xi _3)=0\). Such \(\xi _3\) is unique because of the monotonicity of Q.
Next we show that, if \(\lambda >0\) has been chosen small enough, with the above determined \(\xi _2\) and \(\xi _3\), (2.18) holds. To do this, we consider (2.18) for \(\xi \in (\xi _3,\xi _2)\) and \(\xi \in [\xi _2,0)\) separately.
For \(\xi \in (\xi _3,\xi _2)\), we write \(V_{\varepsilon }=V_{\varepsilon }(\xi )\), \(\gamma =\gamma (\xi -\xi _2)\) and obtain
By (2.20), for \(\xi \in (\xi _3,\xi _2)\),
It follows that
Using (2.9), we see that the right side of (2.21) is nonnegative if the following inequality holds:
We shall show that (2.22) indeed holds provided that \(\lambda >0\) has been chosen small enough. To check this, for \(t=\xi _2-\xi \ge 0\) we define
By Lemma 2.5 below, we can take small \(\lambda \) depending only on \(\varepsilon \) such that \(F(t)>0\) for all \(t\ge 0\). This implies (2.22), and so (2.18) holds for \(\xi \in (\xi _3,\xi _2]\).
For \(\xi \in (\xi _2,0)\), we have \(Q'(\xi )=V_{\varepsilon }(\xi )\). From (2.8), it is straightforward to see that (2.18) holds for \(\xi \in (\xi _2, 0)\). This completes the proof.\(\square \)
Lemma 2.5
Let \(\varepsilon >0\) and \(F:[0,\infty )\rightarrow \mathbb {R}\) be defined by (2.23). Then \(F(t)>0\) for all \(t\ge 0\) as long as \(\lambda >0\) is small enough.
Proof
The argument is similar to [13, Lemma 3.3]. Let \(\kappa :=h-\varepsilon -1\). Note that \(\kappa >0\) since \(h>1\). By direct computations,
where \(\mu >0\) is given in (2.9). By taking
we have \(F(0)>0\), \(F'(0)<0\), \(F'(\infty )=\infty \) and \(F''(t)>0\) for \(t\ge 0\). If follows that F has a unique minimum point \(t=t_{\lambda }\). Consequently, to finish the proof of Lemma 2.5, it suffices to show the following:
By direct calculation, \(F'(t_{\lambda })=0\) implies that
From (2.25), we easily deduce \(t_\lambda \rightarrow \infty \) as \(\lambda \rightarrow 0\), for otherwise, the left hand side of (2.25) is bounded below by a positive constant while the right hand side converges to 0 as \(\lambda \rightarrow 0\) along some sequence. Multiplying \(t_\lambda \) to both sides of (2.25) and we obtain, by a similar consideration, that \(\lambda t_\lambda \) is bounded from above by a positive constant as \(\lambda \rightarrow 0\). It then follows that
which implies \(\lambda t_\lambda \rightarrow 0\) as \(\lambda \rightarrow 0\). We thus obtain
It follows that
We now prove (2.24). Substituting (2.25) into F, we have
Using (2.26) and (2.27), for small \(\lambda >0\),
This completes the proof.\(\square \)
Combining (2.10) and (2.16) we now define
where \(\xi _1>0\) and \(\xi _3<0\) are given in Lemmas 2.3 and 2.3, respectively. We then define
with \(s^{\varepsilon }\in (0, s^*_{\mu _2})\) given in Lemma 2.3. Then clearly \(W_{\varepsilon }\in C(\mathbb {R})\) has compact support \([\xi _3,\xi _1+z(s^{\varepsilon })]\).
We are now ready to describe the conditions in Theorem 1 on the initial functions \(u_0\) and \(v_0\). Since \(s^{\varepsilon }\rightarrow s^*_{\mu _2}>c^*_{\mu _1}\) and \(\xi _0(\varepsilon )\rightarrow \infty \) as \(\varepsilon \rightarrow 0\), where \(\xi _0(\varepsilon )\) is defined in Lemma 2.3, we can fix \(\varepsilon _0>0\) small so that
where N is the space dimension. Our first condition is
(B1): For some \(z_0>0\) and small \(\varepsilon _0>0\) as above,
We note that (B1) implies
Our second condition is
(B2): \(s_1^{0}\ge R^*\sqrt{\frac{d}{r(1-k)}}\), where \(R^*>0\) is defined in Corollary 1.
Since \(\limsup _{t\rightarrow \infty } v(r,t)\le 1\) uniformly in \(r\in [0, s_2(t)]\), it is easy to see that (B2) guarantees \(s_{1,\infty }=\lim _{t\rightarrow \infty } s_1(t)=\infty \) (see also the proof of Theorem 2).
We are now ready to prove Theorem 1, which we restate as
Theorem 3
Suppose that (1.2), (1.7), (B1) and (B2) hold. Then the solution \((u,v,s_1,s_2)\) of (P) satisfies
and for every small \(\epsilon >0\), (1.8), (1.9) hold.
Before giving the proof of Theorem 3, let us first observe how Corollary 1 follows easily from Theorem 3. It suffices to show that assumptions (i) and (ii) in Corollary 1 imply (B1) and (B2). Recall that
Therefore, for fixed \(s_1^0\ge R^*\sqrt{\frac{d}{r(1-k)}}\), there exists \(C_1>0\) large such that
Hence for any given \(u_0\) satisfying (i) of Corollary 1, (B2) and the first inequality in (B1) are satisfied if we take \(z_0\ge C_1\).
From the definition of \(W_{\varepsilon _0}\) we see that
If we take
and for \(x_0\ge C_0\) and \(L\ge C_0\), we let \(z_0:=x_0-\xi _3\), then
and hence \(W_{\varepsilon _0}(r-z_0)=0\) for \(r\not \in [x_0, x_0+L]\). Thus when (ii) in Corollary 1 holds, we have
which is the second inequality in (B1). This proves what we wanted.
Proof of Theorem 3
We break the rather long proof into 4 steps.
Step 1: We show
where
By (2.29), we have
We prove (2.30) by constructing suitable functions \((\overline{U}(r,t), \underline{V}(r,t), l(t), g(t))\) which satisfy certain differential inequalities that enable us to use a comparison argument to relate them to \((u(r,t),v(r,t), s_1(t), s_2(t))\). Set
where \(\widehat{Q}^{\varepsilon _0}\) is defined in (2.28) and \(\xi _1=\xi _1(\varepsilon _0)\) is given in Lemma 2.3. We note that
By the assumption (B1), we have
We now show the wanted differential inequality for \(\overline{U}\):
Using \(\overline{U}_{r}=U_{\varepsilon _0}'<0\), direct computation gives us
When \(\overline{U}>0\), we have \(r\le l(t)\) and so we can divide into two cases: when \(r-l(t)\in (\xi _2,0)\), we have
Hence from (2.35) we see that \(J(r,t)>0\). When \(r-l(t)\in (-l(t),\xi _2)\), by Lemma 2.4 with \(\varepsilon =\varepsilon _0\), we have
Again we obtain from (2.35) that \(J(r,t)>0\). Hence (2.34) holds.
We next show the wanted differential inequality for \(\underline{V}\):
We divide the proof into three parts.
(i) For \(r\in [0,l(t)+\xi _1]\), using \(\underline{V}_r(r,t)=(\widehat{Q}^{\varepsilon _0})'(r-l(t))\ge 0\), Lemma 2.3 and Lemma 2.4 with \(\varepsilon =\varepsilon _0\),
(ii) For \(r\in [l(t)+\xi _1, g(t)-z(s^{{\varepsilon _0}})]\), we have \(\overline{U}\equiv 0\) and \(\underline{V}\equiv \widehat{Q}^{\varepsilon _0}(\xi _1)<1\). So clearly (2.36) holds.
(iii) For \(r\in (g(t)-z(s^{{\varepsilon _0}}), g(t))\), we observe that \(r\ge g(t)-z(s^{{\varepsilon _0}})\ge \xi _1\). Also, by (2.32), we have
Together with the fact that \((q_{s^{\varepsilon _0}})'(r-g(t))<0\) for \(r\in (g(t)-z(s^{{\varepsilon _0}}), g(t))\) and \(t>0\), we have
We have thus proved (2.36).
In order to use the comparison principle to compare \((u,v,s_1, s_2)\) with \((\overline{U},\underline{V}, l, g)\), we note that on the boundary \(r=0\),
Regarding the free boundary conditions, we have
By (2.33), (2.34), (2.36)-(2.39), we can apply the comparison principle ([33, Lemma 3.1] with minor modifications) to deduce that
In particular,
We have thus proved (2.30).
Step 2: We refine the definitions of \((\overline{U}(r,t), \underline{V}(r,t), l(t), g(t))\) in Step 1 to obtain the improved estimates
For any given \(\varepsilon \in (0,\varepsilon _0)\), we redefine \((l,g,\overline{U},\underline{V})\) as
where \(\xi _1=\xi _1(\varepsilon )\) is given in Lemma 2.3 and \(z_1, T_{\varepsilon }\gg 1\) are to be determined later.
We want to show that there exist \(z_1\gg 1\) and \(T_{\varepsilon }\gg 1\) such that
Since \(\limsup _{t\rightarrow \infty }u(r,t)\le 1\) uniformly in r, there exists \(T_{1,\varepsilon }\) such that
By (2.31) and Lemma 2.2, we can find \(0<\nu \ll 1\) and then \(T_{2,\varepsilon }\gg 1\) such that
We now prove (2.41) by making use of (2.42) and (2.44). By the definition of \(\underline{V}(r,t)\), we see that \(\Vert \underline{V}(\cdot ,t)\Vert _{L^{\infty }}<1-\varepsilon \) for all \(t>0\). Also, note that \(\underline{V}(\cdot ,T_{\varepsilon })=W_{\varepsilon }(\cdot -z_1)\) has compact support \([\xi _3+z_1,\xi _1+z(s^\varepsilon )+z_1]\), whose length equals to \(\xi _1-\xi _3+z(s^{\varepsilon })\) which is independent of the choice of \(T_{\varepsilon }\).
Next, we show the following claim: there exist \(z_1\gg 1\) and \(T_{\varepsilon }\gg 1\) such that
Since \(U_{\varepsilon }(-\infty )=1+\varepsilon \), we can find \(T_{3,\varepsilon }\gg 1\) such that
By (2.30), we can find \(T_{4,\varepsilon }\gg 1\) so that
We now take \(z_1:=c_1T_\varepsilon -\xi _3\) with \(T_{\varepsilon }>\max \{T_{1,\varepsilon },T_{2,\varepsilon }, T_{4,\varepsilon }\}\) chosen such that
It follows that
and
Thus we may use (2.42) and (2.44) to obtain
We have thus proved (2.41).
It is also easily seen that, with \(t>0\) replaced by \(t>T_{\varepsilon }\) and \(\varepsilon _0\) replaced by \(\varepsilon \), the inequalities (2.34) and (2.36)–(2.39) still hold. Thus we are able to use the comparison principle as before to deduce
In particular,
Since \(\varepsilon \in (0,\varepsilon _0)\) is arbitrary, taking \(\varepsilon \rightarrow 0\) we obtain (2.40).
Step 3: We prove the following conclusions:
We note that for \(r\in [l(t)+\xi _1, g(t)-z(s^\varepsilon )]\) and \(t>0\),
Thus for any given \(\epsilon >0\) we can choose \(\varepsilon ^*>0\) small enough so that for all \(\varepsilon \in (0, \varepsilon ^*]\),
In view of
and the inequality (2.31), by further shrinking \(\varepsilon ^*\) we may also assume that for all \(\varepsilon \in (0, \varepsilon ^*]\),
Hence for every \(\varepsilon \in (0,\varepsilon ^*]\) we can find \(\tilde{T}_\varepsilon \ge T_\varepsilon \) such that
It follows that
which implies
Next we obtain bounds for \((u,v,s_1,s_2)\) from the other side.
By comparison with an ODE upper solution,
which, combined with (2.47), yields
This proves the second identity in (2.46).
As seen in the proof of Lemma 2.2, we have
Combining this with (2.40), we obtain the first identity in (2.46), namely
We next prove (2.45). Consider the problem (Q) with initial data in (1.3) chosen the following way: \( \hat{u}_0=u_0,\; h_0=s^0_1\) and \(\hat{v}_0\in C^2([0,\infty ))\cap L^{\infty }((0,\infty ))\) satisfies (1.5) and
We denote its unique solution by \((\hat{u}, \hat{v}, h)\). Then by (B2) and Theorem 4.4 in [7], we have \(h_{\infty }=\infty \). Moreover, it follows from Theorem B that
Due to (2.48), we can apply the comparison principle ([33, Lemma 3.1] with minor modifications) to derive
which in particular implies
Moreover, by the definition of \(\psi _\delta \) and \(\underline{h}(t)\) in [13], and the estimate
we easily obtain the following conclusion:
For any given small \(\epsilon >0\), there exists \(\delta ^*>0\) small such that for every \(\delta \in (0, \delta ^*]\), there exists \(T^*_\delta >0\) large so that
It follows that
Hence
By comparison with an ODE upper solution, it is easily seen that
We thus obtain, for any small \(\epsilon >0\),
This proves the second identity in (2.45).
Combining (2.49) and (2.40), we obtain the first identity in (2.45):
Step 4: We complete the proof of Theorem 3 by finally showing that, for any small \(\epsilon >0\),
We prove this by making use of (2.45). Suppose by way of contradiction that (2.50) does not hold. Then for some \(\epsilon _0>0\) small there exist \(\delta _0>0\) and a sequence \(\{(r_k, t_k)\}_{k=1}^\infty \) such that
By passing to a subsequence, we have either (i) \(r_k\rightarrow r^*\in [0, \infty )\) or (ii) \(r_k\rightarrow \infty \) as \(k\rightarrow \infty \).
In case (i) we define
Then
By (2.45), we have \(u_k\rightarrow 1\) in \(L^\infty _{loc}([0,\infty )\times \mathbb {R}^1)\). Since \(v_k(1-v_k-hu_k)\) has an \(L^\infty \) bound that is independent of k, by standard parabolic regularity and a compactness consideration, we may assume, by passing to a subsequence involving a diagonal process, that
and \(v^*\in W^{2,1}_{p, loc}([0,\infty )\times \mathbb {R}^1)\) \((p>1)\) is a solution of
Moreover, \({v^*(r^*,0)}\ge \delta _0\) and due to \(\limsup _{t\rightarrow \infty } v(r,t)\le 1\) we have \(v^*(r,t)\le 1\).
Fix \(R>0\) and let \(\hat{v}(r,t)\) be the unique solution of
By the comparison principle we have, for any \(s>0\),
On the other hand, by the well known properties of logistic type equations, we have
where \(V_R(r)\) is the unique solution to
It follows that
By Lemma 2.1 in [10], we have \(V_R\le V_{R'}\) in \([0, R']\) if \(0<R'<R\). Hence \(V_\infty (r):=\lim _{R\rightarrow \infty } V_R(r)\) exists, and it is easily seen that \(V_\infty \) is a nonnegative solution of
Since \(1-h<0\), by Theorem 2.1 in [10], we have \(V_\infty \equiv 0\). Hence \(\lim _{R\rightarrow \infty } V_R(r)=0\) for every \(r\ge 0\). We may now let \(R\rightarrow \infty \) in (2.51) to obtain \( \delta _0\le 0\). Thus we reach a contradiction in case (i).
In case (ii), \(r_k\rightarrow \infty \) as \(k\rightarrow \infty \), and we define
Since \(r_k\le (c^*_{\mu _1}-\epsilon _0)t_k\), by (2.45) we see that \(u_k(r,t)\rightarrow 1\) in \(L^\infty _{loc}(\mathbb {R}^1\times \mathbb {R}^1)\). Then similarly, by passing to a subsequence, \(v_k(r,t)\rightarrow \tilde{v}^*(r,t)\) in \(C_{loc}^{1+\alpha , \frac{1+\alpha }{2}}(\mathbb {R}^1\times \mathbb {R}^1),\; \alpha \in (0,1)\), and \(\tilde{v}^*\in W^{2,1}_{p, loc}(\mathbb {R}^1\times \mathbb {R}^1)\) \((p>1)\) is a solution of
Moreover, \(\tilde{v}^*(0,0)\ge \delta _0\) and \(\tilde{v}^*(r,t)\le 1\). We may now compare \(\tilde{v}^*\) with the one-dimensional version of \(\hat{v}(r, t)\) used in case (i) to obtain a contradiction. We omit the details as they are just obvious modifications of the arguments in case (i).
As we arrive at a contradiction in both cases (i) and (ii), (2.50) must hold. The proof is now complete.\(\square \)
3 Appendix
This section is divided into three subsections. In Sect. 3.1, we establish the local existence and uniqueness of solutions for a rather general system including (P) as a special case. In Sect. 3.2, we prove the global existence with some additional assumptions on the general system considered in Sect. 3.1, but the resulting system is still much more general than (P). In the final subsection, we give the proof of Theorem 2.
3.1 Local existence and uniqueness
In this subsection, for possible future applications, we show the local existence and uniqueness of the solution to a more general system than (P). Our approach follows that in [15] with suitable changes, and in particular, we will fill in a gap in the argument of [15].
More precisely, we consider the following problem:
where \(r=|x|\), \(\Delta \varphi :=\varphi _{rr}+\frac{(N-1)}{r}\phi _r\), and the initial data satisfies (1.1). We assume that the nonlinear terms f and g satisfy
We have the following local existence and uniqueness result for (3.1).
Theorem 4
Assume (H1) holds and \(\alpha \in (0,1)\). Suppose for some \(M>0\),
Then there exist \(T\in (0,1)\) and \(\widehat{M}>0\) depending only on \(\alpha \), M and the local Lipschitz constants of f and g such that problem (3.1) has a unique solution
satisfying
where \(D^i_{T}:=\{(x,t):0\le x \le s_i(t),\ 0\le t\le T\}\) for \(i=1,2\).
Proof
Firstly, for given \(T\in (0,1)\), we introduce the function spaces
where
Clearly \(s(t)\ge s_i^0\) for \(t\in [0,T]\) if \(s\in \Sigma _T^i\).
For given \((\hat{s}_1,\hat{s}_2)\in \Sigma _T^1\times \Sigma _T^2\), we introduce two corresponding function spaces
We note that \(X^1_T\) and \(X^2_T\) are closed subsets of \(C([0,\infty )\times [0,T])\) under the \(L^\infty ([0,\infty )\times [0,T])\) norm.
Given \((\hat{s}_1,\hat{s}_2)\in \Sigma _T^1\times \Sigma _T^2\) and \((\hat{u}, \hat{v})\in X_T^1\times X_T^2\), we consider the following problem
To solve (3.3) for u, we straighten the boundary \(r=\hat{s}_1(t)\) by the transformation \(R:={r}/{\hat{s}_1(t)}\) and define
Then U satisfies
where
Since
one can apply the standard parabolic \(L^p\) theory and the Sobolev embedding theorem (see [16, 23]) to deduce that (3.4) has a unique solution \( U\in {C^{1+\alpha ,(1+\alpha )/2}([0,1]\times [0,T])}\) with
for some \(C_1\) depending only on \(\alpha \in (0,1)\) and M. It follows that \(u(r,t)=U(\frac{r}{\hat{s}_1(t)}, t)\) satisfies
where \(\tilde{C}_1\) depends only on \(\alpha \) and M, and
Similarly we can solve (3.3) to find a unique \(v\in C^{1+\alpha ,(1+\alpha )/2}(D_T^2)\) satisfying
where \(\tilde{C}_2\) depends only on \(\alpha \) and M, and
We now define a mapping \(\mathcal {G}\) over \(X^1_{T}\times X^2_{T}\) by
and show that \(\mathcal {G}\) has a unique fixed point in \(X^1_{T}\times X^2_{T}\) as long as \(T\in (0,1)\) is sufficiently small, by using the contraction mapping theorem.
For \(R\in [0,1]\) and \(t\in [0,T]\),
It follows that
Similarly,
This implies that \(\mathcal {G}\) maps \(X^1_{T}\times X^2_{T}\) into itself for small \(T\in (0,1)\).
To see that \(\mathcal {G}\) is a contraction mapping, we choose any \((\hat{u}_i,\hat{v}_i)\in X_T^1\times X_T^2\), \(i=1,2\), and set
Then \((\tilde{u},\tilde{v})\) satisfies
By the Lipschitz continuity of f and g, there exists \(C_0>0\) such that for \(r\in [0,\max \{\hat{s}_1(T),\hat{\sigma }_1(T)\}]\) and \(t\in [0, T]\),
We may then repeat the arguments leading to (3.5) and (3.6) to obtain
for some \(C_{2}=C_2(\alpha , M, C_0)\).
If we define
then
for some \(C=C(M)\). Hence from the above estimate for \(\tilde{u}\) and \(\tilde{v}\) we obtain
for some \(C'_{2}=C_2'(\alpha , M, C_0)\). Since \(\tilde{U}(R,0)=\tilde{V}(R,0)\equiv 0\), it follows that
and hence
This implies that \(\mathcal {G}\) is a contraction mapping as long as \(T\in (0,1)\) is sufficiently small. By the contraction mapping theorem, \(\mathcal {G}\) has a unique fixed point in \(X_T^1\times X_T^2\), which we denote by \((\hat{u},\hat{v})\). Furthermore, from (3.5) and (3.6), we have
for some \(\widehat{C}_1=\widehat{C}_1(\alpha , M, C_0)\).
For such \((\hat{u},\hat{v})\), we introduce the mapping
with
Clearly
We shall again apply the contraction mapping theorem to deduce that \(\mathcal {F}\) defined on \(\Sigma _T^1\times \Sigma _T^2\) has a unique fixed point. By (3.7) and (3.8), we see that \(\bar{s}'_i\in C^{{\alpha }/{2}}([0,T])\) with
It follows that
Hence \(\mathcal {F}\) maps \(\Sigma _T^1\times \Sigma _T^2\) into itself as long as \(T\in (0,1)\) is sufficiently small.
To show that \(\mathcal {F}\) is a contraction mapping, we let \((\hat{u}^{s},\hat{v}^{s})\) and \((\hat{u}^{\sigma },\hat{v}^{\sigma })\) be two fixed points of \(\mathcal {G}\) associated with \((\hat{s}_1,\hat{s}_2)\) and \((\hat{\sigma }_1,\hat{\sigma }_2)\in \Sigma _T^1\times \Sigma _T^2\), respectively; and for \(i=1,2\), we denote \(D_T^i\) associated to \((\hat{s}_1,\hat{s}_2)\) and \((\hat{\sigma }_1,\hat{\sigma }_2)\) by, respectively
Let us straighten \(r=\hat{s}_1(t)\) and \(r=\hat{\sigma }_1(t)\), respectively. To do so for \(r=\hat{s}_1(t)\), we define
then \(U^{s}\) satisfies
where
Similarly we set
and find that (3.10) holds with \((U^s, V^s, \hat{s}_1(t))\) replaced by \((U^\sigma , V^\sigma ,\hat{\sigma }_1(t))\) everywhere.
Next we introduce
By some simple computations, P satisfies
where
In view of (3.8),
and hence
From now on, we will depart from the approach of [15] and fill in a gap which occurs in the argument there towards the proof that \(\mathcal {F}\) is a contraction mapping.
It follows from the above identity that
where C depends on \(\mu _1\) and the upper bounds of \(\Vert \hat{s}_1\Vert _{C^{(1+\alpha )/2}([0,T])}\), \(\Vert \hat{\sigma }_1\Vert _{C^{(1+\alpha )/2}([0,T])}\) and \(\Vert U_R^\sigma (1,\cdot )\Vert _{C^{(1+\alpha )/2}([0,T])}\). Hence \(C=C(\alpha , M,C_0)\).
Since \(T\le 1\), clearly
We also have
We thus obtain
Applying the \(L^p\) estimate and the Sobolev embedding theorem to the problem (3.11), we obtain, for some \(p>1\),
for some \(M_4>0\) depending only on \(\alpha \) and M. Due to the \(W^{2,1}_p([0,1]\times [0,T])\) bound for \(U^\sigma \), we hence obtain
for some \(M_5>0\) depending only on \(\alpha \), M and the Lipschitz constant of f.
By the definitions of \(B_1(t),\; B_2(t)\) and F(R, t), we have
and
for some \(C>0\) depending only on M and the Lipschitz constants of f.
We next estimate \(\Vert P\Vert _{C([0,1]\times [0,T])}\) and \(\Vert Q\Vert _{C([0, 1]\times [0,T])}\) by using the estimate in Lemma 2.2 of [15], namely
where
Without loss of generality, we may assume \(\hat{s}_1(t)\le \hat{\sigma }_1(t)\). Then for any \(R\in [0,1]\) and \(t\in [0,T]\), we have
It follows that
For any \(R\in [0,1]\) and \(t\in [0,T]\), we have
We now consider all the possible cases:
(i) If \(R\eta (t)\ge 1\) and \(R\xi (t)\ge 1\), then we immediately obtain
(ii) If \(R\eta (t)<1\) and \(R\xi (t)<1\), assuming without loss of generality \(\hat{s}_2(t)\le \hat{\sigma }_2(t)\), then
(iii) If \(R\eta (t)<1\le R\xi (t)\) and \(R\eta (t)\hat{s}_2(t)\le \hat{\sigma }_2(t)\), then
From \(R\eta (t)<1\le R\xi (t)\) and \(R\eta (t)\hat{s}_2(t)\le \hat{\sigma }_2(t)\) we obtain
Thus in this case we also have
(iv) If \(R\eta (t)<1\le R\xi (t)\) and \(R\eta (t)\hat{s}_2(t)>\hat{\sigma }_2(t)\), then
(v) If \(R\eta (t)\ge 1> R\xi (t)\), we are in a symmetric situation to cases (iii) and (iv) above, so we similarly obtain
Thus in all the possible cases (3.17) always holds. It follows that
We thus obtain from (3.16) that
We may now substitute (3.14), (3.15) and (3.18) into (3.13) to obtain
It thus follows from (3.12) that
Since \(\bar{s}'_1(0)-\bar{\sigma }_1'(0)=0\), this implies
Hence for \(T>0\) sufficiently small we have
with \(\hat{C}_1>0\) depending only on \(\alpha , M\) and the Lipschitz constant of f.
In a similar manner, we can straighten \(r=\hat{s}_2(t)\) and \(r=\hat{\sigma }_2(t)\) to obtain
with \(\hat{C}_2>0\) depending only on \(\alpha , M\) and the Lipschitz constant of g.
Finally, using \(\hat{s}_i(0)=\hat{\sigma }_i(0)=s_i^0\), \(i=1,2\), we see that
Combining (3.19), (3.20) and (3.21), we see that \(\mathcal {F}\) is a contraction mapping as long as \(T>0\) is sufficiently small. Hence \(\mathcal {F}\) has a unique fixed point \((s_1,s_2)\in \Sigma _T^1\times \Sigma _T^2\) for such T.
Let (u, v) be the unique fixed point of \(\mathcal {G}\) in \(X_T^1(s_1, s_2)\times X_T^2(s_1, s_2)\); then it is easily seen that \((u,v,s_1,s_2)\) is the unique solution of (3.1). Furthermore, (3.2) holds because of (3.7) and (3.9). We have now completed the proof of Theorem 4.\(\square \)
By Theorem 4 and the Schauder estimate, we see that the solution of (P) defined for \(t\in [0,T]\) is actually a classical solution.
3.2 Global existence
In this subsection, we show that the unique local solution of (3.1) can be extended to all positive time if the following extra assumption is imposed:
- (H2) :
-
There exists a positive constant K such that \(f(r,t,u,v)\le K(u+v)\) and \(g(r,t,u,v)\le K(u+v)\) for \(r,t, u,v\ge 0\).
Theorem 5
Under the assumptions of Theorem 4 and (H2), problem (3.1) has a unique globally in time solution.
Proof
The proof is similar to that of [7, Theorem 2.4]. For the reader’s convenience, we present a brief proof. Let \([0,T^*)\) be the largest time interval for which the unique solution of (3.1) exists. By Theorem 4, \(T_*>0\). By the strong maximum principle, we see that \(u(r,t)>0\) in \([0,s_1(t))\times [0,T_*)\) and \(v(r,t)>0\) in \([0,s_2(t))\times [0,T_*)\). We will show that \(T_*=\infty \). Aiming for a contradiction, we assume that \(T_*<\infty \). Consider the following ODEs
Take \(M^*>T_*\). Clearly,
By (H2), we can compare (u, v) with (U, V) to obtain
Next, we can use a similar argument as in [6, Lemma 2.2] to derive
for some \(C_2\) independent of \(T_*\). Furthermore, we have
Taking \(\epsilon \in (0,T_*)\), by standard parabolic regularity, there exists \(C_3>0\) depending only on K, \(M^*\), \(C_1\) and \(C_2\) such that
By Theorem 4, there exists \(\tau >0\) depending only on K, \(M^*\) and \(C_i\) (\(i=1,2,3\)) such that the solution of problem (3.1) with initial time \(T_*-\tau /2\) can be extended uniquely to the time \(T_*+\tau /2\), which contradicts the definition of \(T_*\). This completes the proof of Theorem 5. \(\square \)
3.3 Proof of Theorem 2
Proof of Theorem 2
Define
First, following the same lines in [15, Theorem 2] with some minor changes, we can prove the following three results:
-
(i)
If \(s_{1,\infty }\le s_*\), then u vanishes eventually. In this case, v spreads successfully (resp. vanishes eventually) if \(s_{2,\infty }>s^{**}\) (resp. \(s_{2,\infty }\le s^{**}\)),
-
(ii)
If \(s_*<s_{1,\infty }\le s^*\), then u vanishes eventually, and v spreads successfully.
-
(iii)
If \(s_{1,\infty }> s^*\), then u spreads successfully.
Next, we shall show
-
(iv)
If \(s_{1,\infty }> s^*\) and \((\mu _1,\mu _2)\in \mathcal {B}\), then u spreads successfully and v vanishes eventually.
By a simple comparison consideration we see that
By (iii), we see that u spreads successfully and so \(s_{1,\infty }=\infty \). It follows that there exists \(T\gg 1\) such that
This allows us to use a similar argument to that leading to (2.49) but taking T as the initial time to obtain
To show that \(s_{2,\infty }<\infty \) we argue by contradiction and assume \(s_{2,\infty }=\infty \). Since \((\mu _1,\mu _2)\in \mathcal {B}\), from (3.23) and (3.22) we can find \(\tau \gg 1\) and \(\hat{c}\) such that
Then by the same process used in deriving the second identity in (2.45), we have
Also, noting \(h>1\) and \(s_2(t)<\hat{c}t\), there exist \(\hat{\tau }> \tau \) such that
which leads to \(s_{2,\infty }<\infty \) by simple comparison (cf. [15, Theorem 3]). This reaches a contradiction. Hence we have proved \(s_{2,\infty }<\infty \). Finally, using \(s_{2,\infty }<\infty \) we can show \(\lim _{t\rightarrow \infty }\Vert v(\cdot ,t)\Vert _{C([0,s_2(t)])}=0\) (cf. [15, Lemma 3.4]). Hence v vanishes eventually and then (iv) follows.
The conclusions of Theorem 2 follow easily from (i)–(iv). \(\square \)
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Acknowledgements
The authors are grateful to the referee for valuable suggestions on improving the presentation of the paper. YD was supported by the Australian Research Council and CHW was partially supported by the Ministry of Science and Technology of Taiwan under the grant MOST 105-2628-M-024-001-MY2 and National Center for Theoretical Science (NCTS). This research was initiated during the visit of CHW to the University of New England, and he is grateful for the hospitality.
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Communicated by F. H. Lin.
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Du, Y., Wu, CH. Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries. Calc. Var. 57, 52 (2018). https://doi.org/10.1007/s00526-018-1339-5
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DOI: https://doi.org/10.1007/s00526-018-1339-5