Abstract
In this paper, we study the nonlocal dispersal logistic equation
here \(\Omega \subset \mathbb {R}^N\) is a bounded domain, J is a nonnegative dispersal kernel, \(p>1\), \(\lambda \) is a fixed parameter and \(\delta >0\). The coefficients b, q are nonnegative and continuous functions, and q is periodic in t. We are concerned with the asymptotic profiles of positive solutions as \(\delta \rightarrow 0\). We obtain that the temporal degeneracy of q does not make a change of profiles, but the spatial degeneracy of b makes a large change. We find that the sharp profiles are different from the classical reaction–diffusion equations. The investigation in this paper shows that the periodic profile has two different blow-up speeds and the sharp profile is time periodic in domain without spatial degeneracy.
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1 Introduction and main results
Let \(J:\mathbb {R}^N\rightarrow \mathbb {R}\) be a nonnegative continuous function and \(\Omega \subset \mathbb {R}^N\) be a bounded domain. We consider the periodic nonlocal dispersal equation
where \(p>1\) and \(\lambda \) is a real parameter, the coefficient a is nonnegative, T-periodic in t and
represents a nonlocal dispersal operator. It is known that the dispersal operator D and variations of it have been used to model different dispersal phenomena from applications as well as pure mathematics, see [1, 2, 4, 10, 29]. The nonlocal dispersal equation (1.1) arises typically in population dynamics [11, 17, 18]. Let u(y, t) be the density of population at location y at time t, and \(J(x-y)\) be the probability distribution of the population jumping from y to x, then \(\int _{\mathbb {R}^N}J(x-y)u(y,t)dy\) denotes the rate at which individuals are arriving to location x from all other places and \(-u(x,t)=-\int _{\mathbb {R}^N}J(y-x)u(x,t)dy\) is the rate at which they are leaving location x to all other places. Thus Du(x, t) is the dispersal of population and (1.1) describes the change of population density u(x, t) with initial value \(u_I(x)\) and periodic logistic type growth rate. In (1.1), the dispersal takes place in \(\mathbb {R}^N\), but we impose that u vanishes outside \(\bar{\Omega }\), which is called homogeneous nonlocal Dirichlet boundary condition [17]. The operator D is a nonlocal operator since the dispersal of u at location x and time t does not only depend on u, but on all the values of u in a fixed spatial neighborhood of x through the term \(J*u\). There is quite an extensive literature for the study of nonlocal problems recently, among others, the papers [5, 6, 14, 23, 24, 26,27,28].
Since the coefficient a(x, t) may have temporal or spatial degeneracies, the degenerate periodic logistic nonlinearity plays a great role on the dynamical behavior of (1.1), see [27]. In fact, the study of diffusion problems with refuge goes back to the classical works of Fraile et al. [12]. There is quite an extensive literature on the study of degenerate diffusion problems, for example, the papers [8, 12,13,14,15, 20,21,22, 25] and the references therein. In this paper, we shall investigate the influence of degenerate heterogeneous environment on the nonlocal dispersal system (1.1). To this end, we take \(a(x,t)=b(x)q(t)\), where q is T-periodic in t and consider the nonlocal dispersal equation
Throughout this paper, we make the following assumptions on J(x), b(x) and q(t).
- (H1):
\(J\in C(\mathbb {R}^N)\) is nonnegative, symmetric with unit integral and \(J(0)>0\).
- (H2):
\(b\in C(\bar{\Omega })\) and \(q\in C[0,\infty )\) satisfies \(q(t)=q(t+T)\) in \([0,\infty )\) for some \(T>0\).
Our interest here is that the nonlinearity has degeneracies. That is, b(x) or q(t) vanishes in a proper subset. We shall distinguish the following two different cases.
- (A1):
\(b(x)>0\) for all \(x\in \bar{\Omega }\) and \(q(t_q)>0\) for some \(t_q\in [0,T]\).
- (A2):
\(q(t)>0\) for all \(t\in [0,T]\) and \(b(x)=0\) on \(\Omega _0\), while
$$\begin{aligned} b(x)>0 \,\text {for all}\,x\in \bar{\Omega }\setminus \bar{\Omega }_0, \end{aligned}$$here \(\Omega _0\subset \Omega \) is a proper subdomain with positive measure.
The first case is that only the temporal degeneracy exists. We may assume that there exist \(t_0,t_1\in [0,T]\) such that \(q(t)=0\) for \(t\in [t_0,t_1]\) and \(b(x)>0\) for \(x\in \bar{\Omega }\). Then the assumption (A1) holds and the positive solution of the periodic problem (1.2) is well studied, see [23, 27]. Let \(\lambda _P(\Omega )\) be the unique principle eigenvalue of nonlocal equation
we know that (1.2) admits a unique positive solution if and only if \(\lambda >\lambda _P(\Omega )\). If b(x) has a spatial degeneracy, the results are different. If (A2) holds, it follows from [27] that (1.2) admits a unique positive solution if and only if \(\lambda _P(\Omega )<\lambda <\lambda _P(\Omega _0)\).
It is well known from [8, 27] that the dynamical behavior of nonlocal equation (1.2) is different from the classical reaction–diffusion equation
here we assume further that \(\Omega \) is smooth. In order to find the sharp influence of complex environment on the nonlocal dispersal system, we consider the asymptotic profiles of positive periodic solutions. More precisely, we study the perturbed nonlocal dispersal equation
where \(\delta >0\) is a small parameter. In this case, we know that the degeneracy disappears and (1.3) admits a unique positive solution
for \(\lambda >\lambda _P(\Omega )\), see [23, 27]. We want to obtain the sharp behavior of positive solutions when degeneracy appears. So we first establish the asymptotic profiles of positive solutions.
Theorem 1.1
Assume that (A1) holds. Let \(u_\delta (x,t)\) be the unique positive solution of (1.3) for \(\lambda >\lambda _P(\Omega )\) and \(\delta >0\). Then we have
where u(x, t) is the unique positive solution of (1.2).
Theorem 1.2
Assume that (A2) holds. Let \(u_\delta (x,t)\) be the unique positive solution of (1.3) for \(\lambda >\lambda _P(\Omega )\) and \(\delta >0\). Then the following hold.
- (i):
If \(\lambda _P(\Omega )<\lambda <\lambda _P(\Omega _0)\), then
$$\begin{aligned} \lim _{\delta \rightarrow 0+}u_\delta (x,t) =u(x,t) \,\text {uniformly in}\,\bar{\Omega }\times [0,T], \end{aligned}$$where u(x, t) is the unique positive solution of (1.2).
- (ii):
If \(\lambda \ge \lambda _P(\Omega _0)\), then
$$\begin{aligned} \lim _{\delta \rightarrow 0+}u_\delta (x,t)=\infty \text{ uniformly } \text{ in } \bar{\Omega }\times [0,T]. \end{aligned}$$(1.4)
Remark 1.3
If \(b(x)>0\) for \(x\in \bar{\Omega }\) and \(q(t)>0\) for \(t\in [0,T]\), we know that the assumption (A1) still holds. In this case, there is no temporal degeneracy, the conclusion of Theorem 1.1 is also true. We show that only the temporal degeneracy of q(t) does not make a change of the profiles. But if the spatial degeneracy appears, the profiles make a large change. In case of spatial degeneracy, the profiles are also different to the classical reaction–diffusion equation. Let \(\lambda _L(\Omega )\) be the principal eigenvalue of
Then we know from [7, 9, 16, 20] that the classical reaction–diffusion equation
admits a unique positive periodic solution \(u^L_\delta (x,t)\) for \(\lambda >\lambda _L(\Omega )\) and the asymptotic profiles of \(u^L_\delta (x,t)\) with respect to \(\delta \) are well established. If (A1) holds and \(\lambda >\lambda _L(\Omega )\), then
where \(u^L(x,t)\) is the unique positive solution of (1.5) for \(\delta =0\). Meanwhile, if (A2) holds, then we have
for any \(\lambda \in (\lambda _L(\Omega ),\lambda _L(\Omega _0))\) and
for any \(\lambda \ge \lambda _L(\Omega _0)\). In the later case, we know that \(u^L_\delta (x,t)\) is still bounded as \(\delta \rightarrow 0+\) in any compact subset of \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\). However, from (1.4) we obtain that the profiles of nonlocal dispersal equation (1.3) are unbounded in \(\bar{\Omega }\times [0,T]\) as \(\delta \rightarrow 0+\). Thus we know from Theorems 1.1–1.2 that only the temporal degeneracy dose not change the profiles of positive solutions both for nonlocal and classical reaction–diffusion problems. However, the spatial degeneracy makes different changes.
To reveal the complex influence of spatial degeneracy environment on the nonlocal dispersal system (1.3), we investigate the sharp spatial pattern of positive periodic solutions.
Theorem 1.4
Assume that (A2) holds. Let \(u_\delta (x,t)\) be the unique positive solution of (1.3) for \(\lambda >\lambda _P(\Omega )\) and \(\delta >0\). Set \(v_\delta (x,t)=\delta ^{\frac{1}{p-1}}u_\delta (x,t)\), we have the following results.
- (i):
If \(\lambda _P(\Omega )<\lambda \le \lambda _P(\Omega _0)\), then
$$\begin{aligned} \lim _{\delta \rightarrow 0+}v_\delta (x,t) =0 \,\text {uniformly in}\,\bar{\Omega }\times [0,T]. \end{aligned}$$- (ii):
If \(\lambda >\lambda _P(\Omega _0)\), then
$$\begin{aligned} \lim _{\delta \rightarrow 0+}v_\delta (x,t)=\theta (x) \text{ uniformly } \text{ in } \bar{\Omega }_0\times [0,T], \end{aligned}$$(1.6)and
$$\begin{aligned} \lim _{\delta \rightarrow 0+} v_\delta (x,t) =0 \,\text {uniformly in any compact subset of}\, \bar{\Omega }\setminus \bar{\Omega }_0\times [0,T], \end{aligned}$$(1.7)where \(\theta \in C(\bar{\Omega }_0)\) satisfies \(\theta (x)>0\) in \(\bar{\Omega }_0\) and
$$\begin{aligned} \int _{\Omega _0}J(x-y)\theta (y)dy-\theta (x)=-\lambda \theta (x)+\theta ^p(x) \,\text {in}\,\bar{\Omega }_0. \end{aligned}$$(1.8)
Let us note that (1.8) exists a unique positive solution for any \(\lambda >\lambda _P(\Omega _0)\) [14]. Since \(\theta (x)>0\) in \(\bar{\Omega }_0\), the sharp pattern of \(u_\delta (x,t)\) in \(\bar{\Omega }_0\times [0,T]\) is given by (1.6). Due to the effect of nonlocal effect, we know from (1.7) that the pattern is different in \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\). We obtain the sharp profiles of \(u_\delta (x,t)\) in \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\) as follows.
Theorem 1.5
Assume that (A2) holds. Let \(u_\delta (x,t)\) be the unique positive solution of (1.3) for \(\lambda >\lambda _P(\Omega _0)\) and \(\delta >0\). Set \(\omega _\delta (x,t)=\delta ^{\frac{1}{p(p-1)}}u_\delta (x,t)\), we have
and
where
and \(\theta (x)>0\) in \(\bar{\Omega }_0\) is given by (1.8).
Remark 1.6
In the above theorems, we obtain the sharp profiles of positive solutions to the nonlocal dispersal equation (1.3). If (A2) holds, we establish that the sharp profiles in degeneracy domain are different from the domain without degeneracy. In fact, we prove that both the nonlocal effect and the degeneracy of b(x) make the positive periodic solutions of (1.3) blow up, but have different blow-up speeds. Furthermore, we know from (1.9) that the sharp pattern of nonlocal dispersal equation (1.3) is time periodic in domain without degeneracy.
Comparing with the classical reaction–diffusion equation, the sharp pattern for nonlocal dispersal equation is quite different. Our main results reveal the following phenomena for nonlocal dispersal equation (1.3).
(i) The asymptotic profiles are unbounded in the whole domain \(\Omega \).
(ii) The asymptotic profiles have different blow-up speeds, depending on domain \(\Omega _0\).
(iii) The sharp profiles are time independent in degeneracy domain \(\Omega _0\), but time periodic in non-degeneracy domain.
The rest of this paper is organized as follows. In Sect. 2, we investigate the asymptotic profiles. The behavior of principal eigenfunction with respect to parameter is also obtained. Section 3 is devoted to the proofs of sharp profiles.
2 Asymptotic profiles and eigenvalue problems
In this section, we investigate the asymptotic profiles for positive solutions of (1.3). To begin with, we consider the case (A1).
Lemma 2.1
Assume that (A1) holds. Let \(u_\delta (x,t)\) be the unique positive solution of (1.3) for \(\lambda >\lambda _P(\Omega )\) and \(\delta >0\). Then we have
for \(\delta _2\ge \delta _1>0\), here u(x, t) is the unique positive solution of (1.2). Moreover, we have
Proof
Since \(\delta _2\ge \delta _1>0\), we can see that \(u_{\delta _2}(x,t)\) is a lower-solution of (1.3) for \(\delta =\delta _1\). Note that \(u_{\delta _1}(x,t)\) is the unique solution of (1.3) for \(\delta =\delta _1\), then by upper-lower solutions argument (see [2, 27]), we get
Similarly, we have
for \(\delta >0\) and (2.1) holds.
Now by (2.1), we can find a bounded function \(u_0(x,t)\) such that
for \((x,t)\in \bar{\Omega }\times [0,T]\). Thus we know from (1.3) that \(u_0(x,0)=u_0(x,T)\) in \(\bar{\Omega }\) and
for \((x,t)\in \bar{\Omega }\times [0,T]\). Let \(\varepsilon \) be a small parameter, we have
for \((x,t)\in \bar{\Omega }_0\times [0,T]\). Since \(u_0(x,t)\) is uniformly bounded in \(\bar{\Omega }\times [0,T]\), we get
On the other hand, we have
and so
Then by (2.3) we know that \(u_0(x,t)\) is a positive solution of (1.2) and the uniqueness shows that \(u_0(x,t)=u(x,t)\) in \(\bar{\Omega }\times [0,T]\). Thus we obtain (2.2) by Dini’s theorem. \(\square \)
If the spatial degeneracy appears, the case is quite different. To do this, we need to study the periodic nonlocal eigenvalue equation
here \(\mu \ge 0\). By the pioneering work of J. López-Gómez [19], we have the following lemma, one can see [27] for a similar proof.
Lemma 2.2
Assume that (A2) holds. If \(\mu \ge 0\), then (2.4) admits a unique principal eigenvalue \(\lambda _P(\mu ,\Omega )\). Moreover, \(\lambda _P(\mu ,\Omega )\) is strictly increasing with respect to \(\mu \), \(\lambda _P(0,\Omega )=\lambda _P(\Omega )\) and
Now we give the asymptotic behavior of positive eigenfunctions associated with \(\lambda _P(\mu ,\Omega )\), which is a nonlocal version of the classical problem [3].
Theorem 2.3
Assume that (A2) holds. Let \(\phi _\mu (x,t)\) and \(\psi (x)\) be the positive eigenfunctions associated with \(\lambda _P(\mu ,\Omega )\) for \(\mu \ge 0\) and \(\lambda _P(\Omega _0)\) such that
respectively. Then we have
and
Proof
We will prove the main results by the following four steps.
Step 1. We show that \(\phi _\mu (x,t)\) is uniformly bounded in \(\bar{\Omega }\times [0,T]\).
It follows from (2.4) that
where \(q_*=\min _{[0,T]}q(t)\). Denote
since \(\lambda _P(\mu ,\Omega )\le \lambda _P(\Omega _0)<1\), a direct comparison argument gives
for \((x,t)\in \bar{\Omega }\times [0,T]\). This yields
for \(x\in \bar{\Omega }\). Set
again by (2.6) we get
for \((x,t)\in \bar{\Omega }\times [0,T]\).
Step 2. The eigenfunction \(\phi _\mu (x,t)\) in \(\bar{\Omega }_0\times [0,T]\).
Let \(x_1,x_2\in \bar{\Omega }_0\), we denote
Without loss of generality, we assume that \(v(T)>0\). By (2.4) we obtain that
where \(t\in [0,T]\) and
Since \(\lambda _P(\Omega )\le \lambda _P(\mu ,\Omega )<\lambda _P(\Omega _0)<1\) for \(\mu \ge 0\), we get
for \(t\in [0,T]\). But \(v(0)=v(T)\), we have
Meanwhile, we know from (2.8) that
for \(x_1,x_2\in \bar{\Omega }_0\).
On the other hand, for \(x\in \bar{\Omega }_0\) and \(0\le t_1<t_2\le T\), it follows from
that there exist \(t_1^*,t_2^*\in (t_1,t_2)\) such that
Thus we have
for \(x\in \bar{\Omega }_0\) and \(t_1,t_2\in [0,T]\).
Accordingly, subject to a subsequence, we know that there exists \(\hat{\phi }\in C(\bar{\Omega }_0\times [0,T])\) such that
Step 3. The eigenfunction \(\phi _\mu (x,t)\) in \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\).
From (2.6)–(2.7), we know that
Thus we know that
for \((x,t)\in \bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\) and
Step 4. We show \(\hat{\phi }(x,t)=\psi (x)\) in \(\bar{\Omega }_0\times [0,T].\)
In view of (2.9) and (2.10), we get
by the dominated convergence theorem. Then we have
for \((x,t)\in \bar{\Omega }_0\times [0,T]\), here \(\varepsilon \) is a small parameter. Thus we know from (2.7) that
This gives that \(\hat{\phi }(x,\cdot )\in C[0,T]\). Furthermore, we have
and \(\hat{\phi }(x,\cdot )\in C^1([0,T])\) for \(x\in \bar{\Omega }\). Hence,
In view of (2.5), we know that
and the maximum principle shows that
At last, as \(\psi (x)\) is a positive eigenfunction associated with \(\lambda _P(\Omega _0)\), by the uniqueness of principal eigenfunction we obtain
for some constant \(c>0\). It follows from (2.5) and (2.11) that \(c=1\), this also shows that (2.9) holds for the entire sequences. \(\square \)
By a similar argument as in the proof of Theorem 2.3, we have the following lemma.
Lemma 2.4
Assume that (A2) holds. Let \(\phi _\mu (x,t)\) and \(\psi (x)\) be the positive eigenfunctions associated with \(\lambda _P(\mu ,\Omega )\) for \(\mu \ge 0\) and \(\lambda _P(\Omega _0)\) such that
respectively, here
Then we have \(\phi _\mu (x,t)\le 1\) in \(\bar{\Omega }\times [0,T]\),
and
For the time independent nonlocal eigenvalue equation
we know form [27] that (2.12) admits a unique principal eigenvalue \(\sigma _P(\mu ,\Omega )\) for \(\mu \ge 0\) if b(x) exists spatial degeneracy. Then we have the following result.
Corollary 2.5
Assume that \(b\in C(\bar{\Omega })\) is nontrivial, nonnegative and \(\Omega _0=\{x\in \Omega : b(x)=0\}\) has a positive measure. Let \(\phi _\mu (x)\) and \(\psi (x)\) be the positive eigenfunctions associated with \(\sigma _P(\mu ,\Omega )\) for \(\mu \ge 0\) and \(\lambda _P(\Omega _0)\) such that
respectively. Then we have
and
Theorem 1.1 is followed by Lemma 2.1. At the end of this section, we prove Theorem 1.2.
Proof of Theorem 1.2
The conclusion (i) can be proved by the same way as in Lemma 2.1. We only show that claim (ii) is true.
Since \(\lambda _P(\mu ,\Omega )<\lambda _P(\Omega _0)\) for \(\mu >0\) and \(\lambda \ge \lambda _P(\Omega _0)\), we can take \(\delta \) small such that
Let \(\phi _\mu (x,t)\) be a positive eigenfunction associated with \(\lambda _P(\mu ,\Omega )\) and
where M is given in Lemma 2.4. Then we know that \(0<\phi _\mu (x,t)\le 1\) in \(\bar{\Omega }\times [0,T]\) and we can check that \(\mu ^{\frac{1}{p-1}}\phi _\mu (x,t)\) is a lower-solution to (1.3). Since \(u_\delta (x,t)\) is monotone with respect to \(\delta \), by the uniqueness of positive solutions, we get
Letting \(\mu \rightarrow \infty \), again by Lemma 2.4, we have
Now let \(\hat{u}_\delta (x)\) be the unique positive solution of
for \(\lambda \ge \lambda _P(\Omega _0)\), here \(q^*=\max _{[0,T]}q(t)\). Similarly to the above argument, we know that
Since
and
we get
Then by the comparison principle we have
and
\(\square \)
3 Sharp profiles
In this section, we establish the sharp profiles for positive solutions of (1.3). We first give some preliminaries and then prove the main theorems.
3.1 Preliminaries
To begin with, we give some estimates on the profiles of positive solutions to (1.3). Let \(u_\delta \in C^1([0,T];C(\bar{\Omega }))\) be the positive solution of (1.3) for \(\lambda >\lambda _P(\Omega )\) and \(\delta >0\). Denote \(v_\delta (x,t)=\delta ^{\frac{1}{p-1}}u_\delta (x,t)\), then we have
In order to obtain lower and upper bounds for \(v_\delta (x,t)\), we consider the nonlocal dispersal equations
and
It follows from [14, 26] that (3.2) exists a unique positive solution \(\hat{u}\in C(\bar{\Omega })\) for \(\lambda >\lambda _P(\Omega )\) and (3.3) exists a unique positive solution \(\bar{u}\in C(\bar{\Omega })\) for \(\lambda >\lambda _P(\Omega _0)\).
Lemma 3.1
Assume that (A2) holds and \(\delta >0\). Let \(\hat{u}(x)\) be the positive solution of (3.2) and \(\bar{u}(x)\) be the positive solution of (3.3), respectively. Then we have
for \(\lambda >\lambda _P(\Omega )\) and
for \(\lambda >\lambda _P(\Omega _0)\).
Proof
Since b(x) and q(t) are nonnegative, we can see that \(\hat{u}(x)\) is an upper-solution of (3.1). The uniqueness gives that (3.4) holds.
On the other hand, we know that \(v_\delta (x,t)\) satisfies the nonlocal dispersal equation
where
By a simple argument (see the proof of Theorem 5.4 in [27]), we know that \(v_\delta (x,t)\) is the only continuous positive solution of (3.6). But \(\bar{u}(x)\) is a lower-solution to (3.6), we get (3.5).
\(\square \)
Lemma 3.2
Assume that (A2) holds and \(\lambda \ge \lambda _P(\Omega _0)\). Then there exists \(l>0\) which is independent of \(\delta \) such that
for \(\delta >0\).
Proof
Since \(\lambda _P(\Omega _0)\in [0,1)\) and \(v_\delta (x,t)\) is nonnegative, we can see that (3.7) holds for \(\lambda =\lambda _P(\Omega _0)\).
If \(\lambda >\lambda _P(\Omega _0)\), let \(\bar{u}(x)\) be the unique solution of (3.3), we know from (3.5) that
Since \(\bar{u}(x)>0\) in \(\bar{\Omega }_0\), we complete the proof. \(\square \)
3.2 Proof of Theorems 1.4–1.5
In this subsection, we will prove the main theorems.
Proof of Theorems 1.4–1.5
The long discuss is divided into the following steps.
Step 1. The asymptotic profile for \(\lambda _P(\Omega )<\lambda <\lambda _P(\Omega _0)\).
In this case, we know from Theorem 1.2 that
Step 2. The profile \(v_\delta (x,t)\) in \(\bar{\Omega }_0\times [0,T]\) for \(\lambda >\lambda _P(\Omega _0)\).
By (3.4) we can find \(C>0\) which is independent to \(\delta \) such that
For any given \(x_1,x_2\in \bar{\Omega }_0\), we denote
Without loss of generality, we assume that \(\omega (0)=\omega (T)>0\). Since \(\omega (t)\) is continuous in [0, T], we first show that \(\omega (T)\) satisfies
where \(l>0\) is given by (3.7).
If \({\omega }(t)\) is not sign-changing in [0, T], we assume that \({\omega }(t)\ge 0\) for \(t\in [0,T]\). Then we know from (3.1) and (3.7) that
where \(\theta (t)\) is between \(v_\delta (x_2,t)\) and \(v_\delta (x_1,t)\). Thus we get
for \(t\in [0,T]\). But \({\omega }(0)={\omega }(T)\), we know (3.8) holds.
If \({\omega }(t)\) is sign-changing in [0, T]. In this case, (3.8) still true for \({\omega }(T)=0\). If \({\omega }(T)>0\), since \({\omega }\in C([0,T])\) is sign-changing in [0, T], we can see that there exists \(t_*\in (0,T)\) such that \({\omega }(T)\ge {\omega }(t_*)\) and
It follows from (3.1) that
and so
We get \(\omega (T)\) satisfies (3.8). For \(\omega (T)<0\), a similar argument from \(-\omega (T)\) gives that \(\omega (T)\) satisfies (3.8).
Now for any \(x_1,x_2\in \bar{\Omega }_0\), without loss of generality, we assume that
Then by (3.8) and (3.9), we have
for \(t\in [0,T]\).
Note that \(v_\delta (x,t)\) satisfies
we have
for \(t_1,t_2\in [0,T]\).
By a compactness argument from (3.10) and (3.11), subject to a subsequence, we know that there exists \(v\in C(\bar{\Omega }_0\times [0,T])\) such that
Step 3. The profiles \(v_\delta (x,t)\) in \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\) for \(\lambda >\lambda _P(\Omega _0)\).
We consider the nonlocal dispersal equation
where \(q_*=\min _{[0,T]}q(t)\). Let \(u_\delta (x)\) be the unique solution of (3.13) for \(\lambda >\lambda _P(\Omega )\), then the upper-lower solutions argument gives that
for any \((x,t)\in \bar{\Omega }\times [0,T]\), here \(\hat{u}(x)\) is given by (3.4). Since
we get
and
for any \((x,t)\in \bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\).
Step 4. We show that \(v(x,t)=\theta (x)\) in \(\bar{\Omega }_0\times [0,T]\).
In view of (3.12) and (3.15), by dominated convergence theorem, we know that
and \(v(x,0)=v(x,T)\) in \(\bar{\Omega }\). Meanwhile, we know from Lemma 3.1 that v(x, t) is positive and boundned in \(\bar{\Omega }_0\times [0,T]\). Then a simple argument gives that
So we get
Let \(\theta (x)\) be the unique continuous positive solution of (1.8) for \(\lambda >\lambda _P(\Omega _0)\), we can see that \(\theta (x)\) satisfies (3.16). Since the positive solution is unique, we necessarily have
This also implies that (3.12) holds for the entire original sequences.
Step 5. The profiles \(\omega _\delta (x,t)\) in \(\bar{\Omega }\times [0,T]\) for \(\lambda >\lambda _P(\Omega _0)\).
Since \(\omega _\delta (x,t)=\delta ^{\frac{1}{p(p-1)}}u_\delta (x)\), we can see that \(v_\delta (x,t)=\delta ^{\frac{1}{p}}\omega _\delta (x,t)\) and so
Take f to be a smooth T-periodic function such that \(f(0)=f(T)=0\). Multiplying (3.1) by f and integrating in [0, T], we obtain
Letting \(\delta \rightarrow 0+\), we know from (3.12) and (3.14) that
which is uniform in any compact subset of \(\bar{\Omega }\setminus \bar{\Omega }_0\). Due to the arbitrariness of f, we necessary have
for almost everywhere \(t\in [0,T]\) and the convergence is uniform in any compact subset of \(\bar{\Omega }\setminus \bar{\Omega }_0\). Let \(\Omega _c\) be a compact subset of \(\bar{\Omega }\setminus \bar{\Omega }_0\) and denote
for any given \(t\in [0,T]\). Then we know from (3.17) that H(x) is equicontinuous in \(\Omega _c\).
Now we know from (3.15) and (3.17) that there exists \(\delta _0>0\) and such that
and
for \(\delta \le \delta _0\). Since
for \((x,t)\in \bar{\Omega }_c\times [0,T]\), we get
for any \(x\in \bar{\Omega }_c\) and \(t_1,t_2\in [0,T]\). Then by (3.1) we obtain that
for any \(x\in \bar{\Omega }_c\) and \(t_1,t_2\in [0,T]\), here \(\theta _{\delta }(x,t)\) is between \(v_\delta ^p(x,t_1)\) and \(v_\delta ^p(x,t_2)\).
Let us denote
for any given \(x\in \bar{\Omega }_c\). Then by (3.1) we know that
for \(t_1,t_2\in [0,T]\).
Now it follows from (3.17) that
which is uniform in any compact subset of \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\).
At last, since \(v_\delta (x,t)=\delta ^{\frac{1}{p}}\omega _\delta (x,t)\), (3.18) yields that
and so
which is uniform in any compact subset of \(\bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\).
Step 6. The profiles \(v_\delta (x,t)\) in \(\bar{\Omega }\times [0,T]\) for \(\lambda =\lambda _P(\Omega _0)\).
In this case, we know that the only nonnegative solution of (3.16) is \(u(x,t)=0\) in \(\bar{\Omega }_0\times [0,T]\). A similar arguments as in the previous steps, we know that
and
for any \((x,t)\in \bar{\Omega }\setminus \bar{\Omega }_0\times [0,T]\). Note that \(v_\delta (x,t)\) is monotone with respect to \(\delta \), we end our proof by Dini’s theorem. \(\square \)
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Acknowledgements
The author would like to thank the anonymous reviewer for his/her helpful comments. This work was supported by NSF of China (11401277,11731005), FRFCU (lzujbky-2016-99) and China Postdoctoral Science Foundation (2017M611084).
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Sun, JW. Sharp profiles for periodic logistic equation with nonlocal dispersal. Calc. Var. 59, 46 (2020). https://doi.org/10.1007/s00526-020-1710-1
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DOI: https://doi.org/10.1007/s00526-020-1710-1