A class of infinite horizon mean field games on networks

1.   Introduction

In this paper, we study the positive solutions of the periodic-parabolic problem

where $ \Omega $ is a bounded domain of $ \mathbb{R}^N(N\geq1) $ with smooth boundary $ \partial\Omega $, $ \nu $ is the outward normal vector of $ \partial\Omega $, $ \mu > 0 $ and $ p > 1 $ is constant, $ m(x, t)\in C^{\alpha, \frac{\alpha}{2}}(\bar\Omega\times\mathbb{R})\; (0 < \alpha < 1) $ is $ T $-periodic in $ t $, $ k(x, t) $, $ c(x, t)\in C^{\alpha, 1}(\bar\Omega\times\mathbb{R}) $ are positive and $ T $-periodic in $ t $. It is known that the periodic reaction-diffusion equation (1.1) can be accurately used to describe different diffusion phenomena in infectious diseases, microbial growth, and population ecology, see ^[1,2,3,4]. From a biological point of view, $ \Omega $ represents the habitat of species $ u $ and $ \mu k(x, t) $ stands for the diffusion rate, which is time and space dependent. The function $ m(x, t) $ represents the growth rate of species. In this situation, in the subset $ \{(x, t)\in\Omega\times\mathbb{R}:m(x, t) > 0\} $, the species will increase, while in $ \{(x, t)\in\Omega\times\mathbb{R}:m(x, t) < 0\} $, species will decrease. The coefficient $ c(x, t) $ means that environment $ \Omega $ can accommodate species $ u $. There are many interesting conclusions about the study of the reaction-diffusion equation, see ^[5,6,7,8] for the elliptic problems and ^{[9,10,11,12,13,14]} for the periodic problems.

In particular, if $ k(x, t)\equiv k(t) $ for $ x\in\bar\Omega $, problem (1.1) has been well investigated by Hess ^[2], Cantrell and Cosner ^[1]. Let $ \lambda(\mu) $ be the unique principal eigenvalue of the eigenvalue problem

It follows from ^[2,15] that Eq (1.1) has a positive periodic solution $ \theta_\mu(x, t) $ if and only if $ \lambda(\mu) < 0 $. In addition, Dancer and Hess ^[16] and Daners and López-Gómez ^[17] studied the effect of $ \mu $ on the positive periodic solution of Eq (1.1) with various boundary conditions. The most interesting conclusion of ^[11,16,17] is that

here $ \theta(x, t) $ is the maximum nonnegative periodic solution of

However, there is little result on the associated large diffusion and the effect of large diffusion on positive solutions.

Our goal is to study the existence and uniqueness of positive periodic solutions of Eq (1.1) and the asymptotic behavior of positive periodic solutions when the diffusion rate $ \mu $ is large. To this end, let $ \lambda(\mu; m) $ be the principal eigenvalue of

{It is well known that $ \lambda(\mu; m) $ plays a major role in the study of the positive periodic solution of Eq (1.1). The properties of $ \lambda(\mu; m) $ will be established in Section 2. In addition, let $ W^{2, p}_{\nu}(\Omega) = \{u\in W^{2, p}(\Omega):\frac{\partial u}{\partial\nu} = 0\}(N < p < \infty) $. If $ u_0\in W^{2, p}_{\nu}(\Omega) $, it follows from ^[2] that the semilinear initial value problem}

has a unique solution $ U(x, t) = U(x, t;u_0) $ satisfying

Our first result is the existence and uniqueness of positive periodic solutions of Eq (1.1). For simplicity, in the rest of this paper, we use the following notations:

Theorem 1.1. Suppose that $ \int_0^T m_*(t)\, dt > 0 $. Then Eq (1.1) admits a unique positive periodic solution $ \theta_\mu(x, t) $ for all $ \mu > 0 $.

Remark 1.1. By the result of Section 3, we know that there exists a unique positive solution to Eq (1.1) if and only if $ \lambda(\mu; m) < 0 $. In the case $ \int_0^Tm_*(t)\, dt > 0 $, we obtain that $ \lambda(\mu; m) < 0 $.

Next, we study the asymptotic behavior of positive periodic solutions when the diffusion rate is large.

Theorem 1.2. Suppose that

Let $ \theta_\mu(x, t) $ be the unique positive periodic solution of Eq (1.1) for $ \mu > 0 $. Then we have

where $ \omega(t) $ is the unique positive periodic solution of

Remark 1.2. With the approach of local upper-lower solutions developed by Daners and López-Gómez ^[17] in the study of classical periodic-parabolic logistic equations, we can prove that

provided $ \max_{\Omega}\int_0^Tm(x, t)\, dt > 0 $. It also shows that when $ m_*(t) < 0 < \int_0^T\max_{\Omega}m(x, t)\, dt $, populations with small dispersal rates survive, while populations with large dispersal rates perish. This means that a small diffusion rate is a better strategy than a large diffusion rate under appropriate circumstances.

The rest of this paper is arranged as follows: In Section 2, we study the properties of principal eigenvalues for the periodic eigenvalue problems. In Section 3, we mainly study the existence, uniqueness and stability of the positive solution to Eq (1.1). Moreover, we investigate the asymptotic profiles of the positive periodic solution to Eq (1.1) as $ \mu\to\infty $ in Section 4.

2.   Periodic eigenvalue equation

In this section, we consider the principal eigenvalue of Eq (1.2). To this end, we first study the linear initial value problem

where $ 0\leq\tau < T $, $ u_0\in W^{2, p}_{\nu}(\Omega)(N < p < \infty) $ and $ a(x, t)\in C^{\alpha, \frac{\alpha}{2}}(\bar\Omega\times[\tau, \infty)) $. It is well known that there is a one-to-one correspondence between Eq (2.1) and the evolution operator $ U_\mu(t, \tau) $. Then we can define that $ u(x, t) = U_\mu(t, \tau)u_0 $ is the solution of Eq (2.1). For simplicity, let $ X = L^p(\Omega)(N < p < \infty) $, $ X_1 = W^{2, p}_{\nu}(\Omega) $ and

Inspired by the classical works of Hess ^[2], we first give some important results of Eq (2.1), which will be used in the rest of this paper.

Lemma 2.1. If $ u_0\in X $ is positive, then $ U_\mu(t, \tau)u_0 > 0 $ in $ C^1_{\nu}(\bar\Omega) $ for $ 0\le\tau < t\leq T $.

Proof. Note that $ X_1 $ is compactly embedded in $ X $. The operator $ U_\mu(t, \tau)/_{X_1}:X_1\to X_1 $ can be continuously extended to the positive operator $ U_\mu(t, \tau)\in\mathcal{L}(X, X_1) $. Thus $ U_\mu(t, \tau)u_0\geq0 $ in $ X_1 $. Since $ s\mapsto U_\mu(s, \tau)u_0 $ is continuous from $ [\tau, T] $ to $ X_1 $ and $ U_\mu(\tau, \tau)u_0 = u_0\neq0 $, we can get that $ U_\mu(s, \tau)u_0 > 0 $ in $ X_1 $ as $ s > \tau $ goes to $ \tau $. In addition, we have

for $ \tau < s < t $. Thus it can be obtained that $ U_\mu(t, \tau)u_0 > 0 $ for $ 0\le\tau < t\leq T $.

We now study the periodic-parabolic eigenvalue problem

If there is a nontrivial solution $ u(x, t) $ of Eq (2.2), then $ \lambda(\mu; a) $ is called the eigenvalue. In particular, if $ u(x, t) $ is positive, then $ \lambda(\mu; a) $ is the principal eigenvalue.

Theorem 2.1. Let $ K_\mu: = U_\mu(T, 0) $ and $ r $ be the spectral radius of $ K_\mu $. Then $ r $ is the principal eigenvalue of $ K_\mu $ with positive eigenfunction $ u_0 $ if and only if $ \lambda(\mu; a) = -\frac{1}{T}\ln r $ is the principal eigenvalue of Eq (2.2) with positive eigenfunction $ u(x, t) = e^{\lambda(\mu; a) t}U_\mu(t, 0)u_0 $.

Proof. It can be proved by the similar arguments as in [2, Proposition 14.4]. For the completeness, we provide a proof in the following. Suppose that $ r $ is the principal eigenvalue of $ K_\mu $ with positive eigenfunction $ u_0\in X_1 $. Let $ u(x, t) = e^{\lambda(\mu; a) t}U_\mu(t, 0)u_0 $. Then $ u(x, t) $ satisfies

According to the regularity results, we have $ u(x, t)\in C^{2+\alpha, 1+\frac{\alpha}{2}}(\bar\Omega\times\mathbb{R}) $. This means that $ \mu = -\frac{1}{T}\ln r $ is the principal eigenvalue of Eq (2.2), while $ u(x, t) = e^{\lambda(\mu; a) t}U_\mu(t, 0)u_0 $ is the corresponding positive eigenfunction.

On the contrary, suppose that $ \lambda(\mu; a) = -\frac{1}{T}\ln r $ is the eigenvalue of Eq (2.2) with positive eigenfunction $ u(x, t) $. Set $ v(x, t) = e^{-\lambda(\mu; a) t}u(x, t) $. Then $ v(x, t) $ is the solution of

Thus, we obtain $ v(x, t) = U_\mu(t, 0)u_0 $ for $ 0\leq t\leq T $ and $ u_0\in X_1 $ is positive. Hence,

It follows from Krein-Rutman theorem that $ e^{-\lambda(\mu; a)T} = r $.

Remark 2.1. For $ \tau < t $, it follows that $ U_\mu(t, \tau) $ is a compact and strongly positive operator on $ X_1 $. Moreover, by Krein-Rutman theorem, we obtain $ r > 0 $, and $ r $ is the unique principal eigenvalue of $ K_\mu $. This implies that Eq (2.2) has the unique principal eigenvalue $ \lambda(\mu; a) $ for any $ \mu > 0 $.

Lemma 2.2. Let $ a_1(x, t) $, $ a(x, t)\in F $ satisfy

Then $ \lambda(\mu; a_2) < \lambda(\mu; a_1) $ for any $ \mu > 0 $.

Proof. Assume that there exists $ \mu_1 > 0 $ such that $ \lambda(\mu_1;a_2)\geq\lambda(\mu_1;a_1) $. Let $ u_1(x, t) $ and $ u_2(x, t) $ be corresponding positive eigenfunctions, chosen in such a way that

Then $ \omega(x, t) = u_2(x, t)-u_1(x, t) $ satisfies

Set $ \phi(x, t) = e^{-\lambda(\mu_1;a_1)t}\omega(x, t) $, then we have

Thus, for any $ x\in\Omega $, we can obtain

Hence, we obtain

Note that $ \omega(x, 0) > 0 $, it follows from [2, Theorem 7.3] that

where $ r_{\mu_1} $ is the principal eigenvalue of $ K_{\mu_1} $. This is a contradiction.

Lemma 2.3. Suppose that for any $ n\in\mathbb{N} $, $ a_n(x, t)\in F $ satisfies

Then for fixed $ \mu > 0 $, we have

Proof. For any given $ \varepsilon > 0 $, there exists $ n_\varepsilon\in\mathbb{N} $ such that for any $ n > n_\varepsilon $, there holds

Notice that $ \lambda(\mu; a\pm\varepsilon) = \lambda(\mu; a)\mp\varepsilon $. From Lemma 2.2, we have

for any $ n > n_\varepsilon $.

Lemma 2.4. Let $ \lambda(\mu; a) $ be the principal eigenvalue of Eq (2.2) for $ \mu > 0 $. Then we have

here $ a_*(t) = \int_{\Omega}\frac{a(x, t)}{k(x, t)}\, dx $.

Proof. First, we consider the case

Let $ \varphi(x, t) $ be the positive eigenfunction corresponding to the principal eigenvalue $ \lambda(\mu; a) $. Taking $ \alpha > 0 $ satisfies

Then $ \varphi_\alpha: = \alpha\varphi(x, t) $ is also the principal eigenfunction of Eq (2.2). It is easy to obtain

This implies that Eq (3.2) holds.

Next, we consider the case of

We can find smooth $ T $-periodic functions $ \{k_n(x, t)\} $ such that

and

It follows from Lemma 2.3 that

where $ \lambda_n(\mu; a) $ is the principal eigenvalue of Eq (2.2) with $ k(x, t) $ replaced by $ k_n(x, t) $. It is clear from Eq (2.4) that

Letting $ n\to\infty $, we have Eq (3.2).

Remark 2.2. In Eq (3.2), we obtain upper estimates for the principal eigenvalue of the Neumann problem Eq (2.2). Indeed, let $ \lambda_D $ be the principal eigenvalue of the eigenvalue problem

By a similar way as in ^[2], we can show

for any $ \mu > 0 $.

3.   Positive solutions of periodic-parabolic equation

In this section, we study the existence and uniqueness of positive solutions of Eq (1.1). First, we show that if Eq (1.2) has negative principal eigenvalues, then Eq (1.1) has a unique positive solution. To this end, we recall the upper-lower solutions of Eq (1.1). For the sake of convenience, let

Definition 3.1. The continuous function $ \bar u(x, t) $ is called the upper-solution of Eq (1.1), if

is satisfied.

The definition of the lower-solution is similar. We then can prove the following result, see ^[2,4,5,15].

Theorem 3.1. Suppose that $ \bar u(x, t) $, $ \underline u(x, t) $ are a pair of ordered bounded upper-lower solutions of Eq (1.1). Then Eq (1.1) has a unique positive periodic solution $ \theta_\mu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) $ that satisfies

Proof. Let

Then there exists a constant $ K > 0 $ such that

for any $ (x, t, u_i)\in\bar Q_T\times[\underline u(x, t), \bar u(x, t)] $, $ i = 1, 2 $. It follows from $ L^p $ theory that for any $ u\in C^{\alpha, \frac{\alpha}{2}}(\bar Q_T) $ satisfying $ [\underline u(x, t), \bar u(x, t)] $, the linear initial value problem

admits a unique solution $ v $. Thus, a nonlinear operator $ v = \mathcal{F}u $ is defined. We will prove that there is a fixed point for $ \mathcal{F} $ in four steps.

$ \bf{Step\; 1.} $ In this step, we prove that if $ \underline u\leq u_1\leq u_2\leq\bar u $, then $ \underline u\leq v_1 = \mathcal{F} u_1\leq v_2 = \mathcal{F} u_2\leq\bar u $.

Take $ \omega_1 = v_2-v_1 $, then $ \omega_1 $ satisfies

By the comparison principle, we obtain $ \omega_1\geq0 $. This implies $ \mathcal{F}u_2\geq\mathcal{F}u_1 $. Similarly, let $ \omega_2 = v_1-\underline u $, then $ \omega_2 $ satisfies

Thus, $ \underline u\leq v_1 $. Similarly, $ v_2\leq\bar u $.

$ \bf{Step\; 2.} $ In this step, we construct a convergent monotone sequence.

The iterative sequences $ \{u_n\} $ and $ \{v_n\} $ are constructed as follows:

Since $ \underline u\leq\bar u $ and $ \mathcal{F} $ is monotonically non-decreasing, then

Similarly, we obtain

And since $ \underline u\leq u_1\leq\bar u $,

By induction, we have $ u_{n+1}\leq u_n $. In the same way, we obtain $ v_n\leq v_{n+1} $. Thus, we have

This also implies that $ \{u_n\} $ and $ \{v_n\} $ are monotonically bounded sequences, so there are $ u_0(x, t) $ and $ v_0(x, t) $ such that

Thus

$ \bf{Step\; 3.} $ In this step, we prove that $ u_0(x, t) $ and $ v_0(x, t) $ are solutions of Eq (1.1).

Take $ E = W^{2, 1}_p(Q_T)(p > n+2) $. First, we prove that $ \mathcal{F}:D\to C(\bar Q_T) $ is a compact operator, where

For $ u_1 $, $ u_2\in E $, let $ v_1 = \mathcal{F}u_1 $ and $ v_2 = \mathcal{F}u_2 $, then $ \omega_3 = v_2-v_1 $ satisfies

By the $ L^p $ estimate and embedding theorem, it follows that

here $ C $ and $ C_1 $ are positive constants. Thus $ \mathcal{F}:D\to C(\bar Q_T) $ is continuous. It is known from the embedding theorem that if $ u $ is bounded in $ W^{2, 1}_p(Q_T) $, then $ \mathcal{F}u $ is bounded in $ C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) $. This means that $ \mathcal{F}:D\to C(\bar Q_T) $ is a compact operator.

Since $ u_n $ is bounded, $ u_n = \mathcal{F}u_{n-1} $ has a convergent subsequence in $ C(\bar Q_T) $. By the monotonicity of $ u_n $,

Thus $ u_0(x, t) $ is the solution of Eq (1.1) in $ W^{2, 1}_p(Q_T) $. The embedding theorem is used again to get $ u_0(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) $. In the same way, we get that $ v_0(x, t) $ is the classical solution of Eq (1.1).

$ \bf{Step\; 4.} $ In this step, we prove the uniqueness and periodicity of the solution of Eq (1.1).

Since $ k(x, t) $, $ m(x, t) $ and $ c(x, t) $ are periodic about $ t $, then $ \tau(x, t) = u_0(x, t+T)-u_0(x, t) $ satisfies

here $ \tilde{u}(x, t) $ is between $ u_0(x, t+T) $ and $ u_0(x, t) $. It is well known that the solution of Eq (3.1) is unique, thus $ u_0(x, t+T)\equiv u_0(x, t) $ in $ \bar Q_T $.

The uniqueness of the solution is based on the results of ^[2,4] and can also be found in recent research results ^[11,15]. Assume that $ v_1 $ and $ v_2 $ are two positive periodic solutions of Eq (1.1). We first prove that there exists a large constant $ M > 1 $ such that

Indeed, it is clear that there exists $ M_1 > 1 $ such that

This implies $ v_2(x, 0)\not\equiv M_1v_1(x, 0) $ on $ \bar\Omega $. Let $ \eta(x, t): = M_1v_1(x, t)-v_2(x, t) $, then $ \eta(x, t) $ satisfies

where $ \varsigma(x, t) $ is lying between $ M_1v_1(x, t) $ and $ v_2(x, t) $. Notice that $ \frac{\partial\eta}{\partial\nu} = 0 $ on $ Q_1 $. By the maximum principle, we have $ \eta > 0 $ in $ \bar Q_T $. Similarly, we can obtain that there exists $ M_2 > 0 $ such that $ v_1 < M_2v_2 $ in $ \bar Q_T $. Take $ M = \max\{M_1, M_2\} $, then we have

We know that $ Mv_1(x, t) $ and $ M^{-1}v_1(x, t) $ are a pair of upper-lower solutions of Eq (1.1). According to the second step, Eq (1.1) has a minimum solution $ u_* $ and a maximum solution $ u^* $, which satisfies $ u_*\leq v\leq u^* $ in $ \bar Q_T $ for all solution $ v $ satisfying $ M^{-1}v_1\leq v\leq Mv_1 $. Thus, we obtain $ u_*\leq v_i\leq u^* $ for $ i = 1, 2 $. Hence, $ u_*\equiv u^* $ implies the uniqueness of the solution to Eq (1.1). Set

It is clear that $ \alpha_{*}\geq1 $. Note that if $ \alpha_{*} = 1 $, then $ u_*(x, t)\equiv u^*(x, t) $ in $ \bar Q_T $. Assume that $ \alpha_{*} > 1 $. Let $ \pi(x, t) = \alpha_{*} u_*-u^* $. It is known from the maximum principle that $ \pi(x, t) > 0 $ in $ \bar Q_T $. On the other hand, we know that

for some small $ \alpha_1 > 0 $. We can use the previous method to prove the existence of $ M $ to show that

Then we have $ u^*(x, t)\leq (\alpha_*-\alpha) u_*(x, t) $. This is in contradiction with the definition of $ \alpha_* $. Thus, we obtain $ \alpha_{*} = 1 $. The uniqueness is proved.

Lemma 3.1. If $ \lambda(\mu; m) < 0 $, then Eq (1.1) admits a unique positive periodic solution $ \theta_\mu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) $. Moreover, $ \theta_\mu(x, t) $ is globally asymptotically stable.

Proof. Let $ \theta(x, t) $ be a principal eigenfunction of Eq (1.2) normalized by $ \|\theta(x, t)\|_{C(\bar Q_T)} = 1 $. Then $ \underline{u} = \varepsilon\theta(x, t) $ is a lower-solution of Eq (1.1) for any

Take

Then we have $ \bar u = M\theta(x, t) $ is an upper-solution of Eq (1.1). From Theorem 3.1, we get that Eq (1.1) has a unique positive solution $ \theta_\mu(x, t) $.

Since $ \theta_\mu(x, t) $ is the solution of Eq (1.1), then $ \lambda(\mu; m-c\theta_\mu^{p-1}) = 0 $. Let $ \lambda_1 $ be the eigenvalue of the linear problem

Due to

for $ u > 0 $. Then $ \lambda_1 > 0 $. It follows from Theorem 2.1 that $ r < 1 $. Thus, $ \theta_\mu(x, t) $ is locally asymptotically stable. In addition, we can choose $ \varepsilon $ small enough and $ M $ large enough such that $ \varepsilon\theta(x, t) $ and $ M\theta(x, t) $ are a pair of ordered bounded upper-lower solutions of Eq (1.1). Then we know that $ \theta_\mu(x, t) $ is globally asymptotically stable by the standard iteration argument as in ^[2].

Lemma 3.2. If (1.1) has a positive periodic solution, then $ \lambda(\mu; m) < 0 $.

Proof. Let $ \theta_\mu(x, t) $ be a positive periodic solution of Eq (1.1). Thanks to ^[2], we can get that Eq (1.1) is equivalent to

Notice that $ \theta_\mu(x, t) > 0 $. We now apply [2, Theorem 7.3] to obtain

Thus, $ \lambda(\mu; m) < 0 $.

Proposition 3.1. If $ \int_0^Tm_*(t)\, dt > 0 $, then Eq (1.1) admits a unique positive periodic solution $ \theta_\mu(x, t) $ for all $ \mu > 0 $.

Proof. According to Lemma 2.4, we know that

Due to $ \int_0^Tm_*(t)\, dt > 0 $, $ \lambda(\mu; m) < 0 $. This together with Lemma 3.1 implies that Eq (1.1) admits a unique positive periodic solution for all $ \mu > 0 $.

4.   Positive solutions with large diffusion rate

In this section, we study the asymptotic behavior of the positive periodic solution of Eq (1.1) when the diffusion rate is large. Here we use regularity estimates together with the perturbation technique to prove our main result. To do this, we first consider the perturbation equation

where the parameter $ \varepsilon > 0 $.

Lemma 4.1. Assume that Eq (1.3) holds. Then Eq (4.1) has a positive periodic solution $ \theta^{\varepsilon}_{\mu}(x, t) $ for $ \mu > 0 $, provided $ \varepsilon > 0 $ is small. Moreover, we can find $ \mu_1 > 0 $ such that

for $ \mu\geq\mu_1 $.

Proof. Through a similar argument as in Theorem 3.1, we know the existence of the positive solution $ \theta^{\varepsilon}_{\mu}(x, t) $ to Eq (4.1). We only need to prove Eq (4.2). Let $ \sigma(x, t) = \theta^{\varepsilon}_{\mu}(x, \frac{t}{\mu}) $. Then $ \sigma(x, t) $ satisfies

It is known from the $ L^p $ estimate that there exists $ \mu_1 > 0 $ such that $ \sigma(x, t) $ is bounded in $ W^{2, 1}_p(\Omega\times(0, \mu T]) $ for any $ \mu > \mu_1 $. It then follows that $ \theta^{\varepsilon}_{\mu}(x, t) $ is bounded in $ W^{2, 1}_p(Q_T) $ for any $ \mu > \mu_1 $. Then, taking $ p $ large enough, we know from the embedding theorem that $ \theta^{\varepsilon}_{\mu}(x, t) $ is compact in $ C^{1, \frac{1}{2}}(\bar Q_T) $. Thus there is a subsequence, still denoted by $ \theta_\mu^{\varepsilon}(x, t) $, such that

for some nonnegative periodic function $ \nu(x, t)\in C(\bar Q_T) $. It follows from the argument of Lemma 3.1 that $ \varepsilon\theta(x, t) $ is a lower-solution of Eq (4.1). Thus we have $ \nu(x, t) > 0 $ for $ (x, t)\in\bar\Omega\times[0, T] $. Since $ \theta^{\varepsilon}_{\mu}(x, t) $ is bounded and Eq (4.3), $ \nu $ satisfies

It is easy to obtain

By standard parabolic regularity, we know that $ \nu(x, t)\in C^{1+\alpha, (1+\alpha)/2}(\bar Q_T) $. The uniqueness of the solution means that Eq (4.2) holds.

At the end of this section, we prove Theorem 1.2.

Proof of Theorem 1.2. We divide the proof into the following three steps.

$ \bf{Step\; 1.} $ In this step, we prove that $ \theta_\mu(x, t) $ has a convergent subsequence as $ \mu\to\infty $.

It follows from a similar argument to Lemma 4.1 that there exists $ \mu_1 > 0 $ such that $ \theta_\mu(x, t) $ is compact in $ C^{1, \frac{1}{2}}(\bar Q_T) $ for any $ \mu > \mu_1 $. Thus, by passing to a subsequence, there is a nonnegative periodic function $ \theta\in C(\bar Q_T) $ such that

$ \bf{Step\; 2.} $ In this step, we show that $ \theta(x, t) $ is independent of $ t $.

Let $ f(t) $ be a smooth $ T $-periodic function, then we have

By dividing $ \mu $ and making $ \mu\to\infty $, we have

Since $ f(t) $ is arbitrary, we obtain

Then we derive

Thus we have $ \theta(x, t)\equiv\theta(t) $ for $ x\in\bar\Omega $.

$ \bf{Step\; 3.} $ In this step, we show that $ \theta(t) = \omega(t) $ in $ [0, T] $.

First, we assert that $ \theta(t)\in C^{1}((0, \infty)) $. Indeed, it is easy to obtain from Eq (1.1) that

Then we have

Taking $ \mu\to\infty $, we obtain

Thus, we derive

Hence,

for $ t > 0 $. Thus $ \theta(t)\in C^{1}((0, \infty)) $ holds.

We then prove that $ \theta(t) $ satisfies Eq (1.5). It is obvious from Eq (1.1) that

Similarly, suppose that $ f(t) $ is a smooth $ T $-periodic function. Multiplying $ f(t) $ on both sides of Eq (4.4) and integrating over $ [0, T] $ gives

Letting $ \mu\to\infty $, we know

This implies

By the arbitrary of $ f(t) $, it follows that

Thus, we have

Finally, we prove that $ \theta(t) > 0 $ in $ t\in[0, T] $. We define $ \theta^{\varepsilon}_{\mu}(x, t) $ as the unique positive periodic solution of Eq (4.1) for small $ \varepsilon > 0 $ and large $ \mu $. Similarly to the previous argument, it can be shown that

where $ \omega^{\varepsilon}(t) $ satisfies

Since

we can obtain that Eq (1.5) admits a unique periodic positive solution $ \omega(t) $. Note that $ \omega(t) $ is a lower solution of Eq (4.6). Then there exists a unique positive periodic solution $ \omega^{\varepsilon}(t) $ to Eq (4.6). In addition, $ \omega^{\varepsilon}(t) $ is monotonically increasing about $ \varepsilon $, and $ \omega_{\varepsilon}(t)\geq\omega(t) > 0 $. We obtain that there exists a positive continuous function $ \omega_{0}(t) $ such that

The uniqueness of the positive solution of Eq (4.6) implies that

It follows from Lemma 4.1 that

This means that $ \theta(t) $ is positive, together with (4.4)–(4.6). Thus, we must have

This also implies that

holds for the entire sequence.

5.   Conclusions

We consider the positive solutions of the periodic-parabolic logistic equation with indefinite weight function and nonhomogeneous diffusion coefficient. When the dispersal rate is small, we can obtain a similar result as in the homogeneous diffusion coefficient. Here we are interested in the case of large diffusion coefficient with nonhomogeneous diffusion coefficient.

In Theorem 1.1, we obtain the condition of $ m(x, t) $ to guarantee a positive periodic solution for all $ \mu > 0 $. Then we investigate the effect of large $ \mu $ on the positive solution and establish that the limiting profile is determined by the positive solution of Eq (1.5). More precisely, we prove that the positive periodic solution tends to the unique positive solution of the corresponding non-autonomous logistic equation when the diffusion rate is large.

Author contributions

M. Fan was responsible for writing the original draft. J. Sun handled the review and supervision.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the anonymous reviewers for the careful reading and several valuable comments to revise the paper. Fan was supported by Gansu postgraduate scientifc research (20230XZX-055) and Sun was supported by NSF of China (12371170) and NSF of Gansu Province of China (21JR7RA535, 21JR7RA537).

Conflict of interest

The authors declare there is no conflict of interest.