A mechanical ventilator is an important medical equipment that assists patients who have breathing difficulties. In recent times a huge percentage of COVID-19 infected patients suffered from respiratory system failure. In order to ensure the abundant availability of mechanical ventilators during COVID-19 pandemic, most of the manufacturers around the globe utilized open source designs. Patients safety is of utmost importance while using mechanical ventilators for assisting them in breathing. Closed loop feedback control system plays vital role in ensuring the stability and reliability of dynamical systems such as mechanical ventilators. Ideal characteristics of mechanical ventilators include safety of patients, reliability, quick and smooth air pressure buildup and release.Unfortunately most of the open source designs and mechanical ventilator units with classical control loops cannot achieve the above mentioned ideal characteristics under system uncertainties. This article proposes a cascaded approach to formulate robust control system for regulating the states of ventilator unit using blower model reduction techniques. Model reduction allows to cascade the blower dynamics in the main controller design for airway pressure. The proposed controller is derived based on both integer and non integer calculus and the stability of the closed loop is ensured using Lyapunov theorems. The effectiveness of the proposed control method is demonstrated using extensive numerical simulations.
Citation: Nasim Ullah, Al-sharef Mohammad. Cascaded robust control of mechanical ventilator using fractional order sliding mode control[J]. Mathematical Biosciences and Engineering, 2022, 19(2): 1332-1354. doi: 10.3934/mbe.2022061
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A mechanical ventilator is an important medical equipment that assists patients who have breathing difficulties. In recent times a huge percentage of COVID-19 infected patients suffered from respiratory system failure. In order to ensure the abundant availability of mechanical ventilators during COVID-19 pandemic, most of the manufacturers around the globe utilized open source designs. Patients safety is of utmost importance while using mechanical ventilators for assisting them in breathing. Closed loop feedback control system plays vital role in ensuring the stability and reliability of dynamical systems such as mechanical ventilators. Ideal characteristics of mechanical ventilators include safety of patients, reliability, quick and smooth air pressure buildup and release.Unfortunately most of the open source designs and mechanical ventilator units with classical control loops cannot achieve the above mentioned ideal characteristics under system uncertainties. This article proposes a cascaded approach to formulate robust control system for regulating the states of ventilator unit using blower model reduction techniques. Model reduction allows to cascade the blower dynamics in the main controller design for airway pressure. The proposed controller is derived based on both integer and non integer calculus and the stability of the closed loop is ensured using Lyapunov theorems. The effectiveness of the proposed control method is demonstrated using extensive numerical simulations.
In this paper, we study the positive solutions of the periodic-parabolic problem
{ut=μk(x,t)Δu+m(x,t)u−c(x,t)up, in Ω×R,∂u∂ν=0, on ∂Ω×R,u(x,0)=u(x,T), in Ω, | (1.1) |
where Ω is a bounded domain of RN(N≥1) with smooth boundary ∂Ω, ν is the outward normal vector of ∂Ω, μ>0 and p>1 is constant, m(x,t)∈Cα,α2(ˉΩ×R)(0<α<1) is T-periodic in t, k(x,t), c(x,t)∈Cα,1(ˉΩ×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, Ω represents the habitat of species u and μk(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)∈Ω×R:m(x,t)>0}, the species will increase, while in {(x,t)∈Ω×R:m(x,t)<0}, species will decrease. The coefficient c(x,t) means that environment Ω 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)≡k(t) for x∈ˉΩ, problem (1.1) has been well investigated by Hess [2], Cantrell and Cosner [1]. Let λ(μ) be the unique principal eigenvalue of the eigenvalue problem
{ut−μk(t)Δu−m(x,t)u=λ(μ)u, in Ω×R,∂u∂ν=0, on ∂Ω×R,u(x,0)=u(x,T), in Ω. |
It follows from [2,15] that Eq (1.1) has a positive periodic solution θμ(x,t) if and only if λ(μ)<0. In addition, Dancer and Hess [16] and Daners and López-Gómez [17] studied the effect of μ on the positive periodic solution of Eq (1.1) with various boundary conditions. The most interesting conclusion of [11,16,17] is that
limμ→0+θμ(x,t)=θ(x,t) locally uniformly in Ω×[0,T], |
here θ(x,t) is the maximum nonnegative periodic solution of
{ut=m(x,t)u−c(x,t)up,t∈R,u(x,0)=u(x,T). |
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 μ is large. To this end, let λ(μ;m) be the principal eigenvalue of
{ut−μk(x,t)Δu−m(x,t)u=λ(μ;m)u, in Ω×R,∂u∂ν=0, on ∂Ω×R,u(x,0)=u(x,T), in Ω. | (1.2) |
{It is well known that λ(μ;m) plays a major role in the study of the positive periodic solution of Eq (1.1). The properties of λ(μ;m) will be established in Section 2. In addition, let W2,pν(Ω)={u∈W2,p(Ω):∂u∂ν=0}(N<p<∞). If u0∈W2,pν(Ω), it follows from [2] that the semilinear initial value problem}
{ut=μk(x,t)Δu+m(x,t)u−c(x,t)up, in Ω×R,∂u∂ν=0, on ∂Ω×R,u(x,0)=u0(x), in Ω, |
has a unique solution U(x,t)=U(x,t;u0) satisfying
U(x,t)∈C1+α,1+α2(ˉΩ×[0,T])∩C2+α,1+α2(ˉΩ×(0,T]). |
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:
k∗(t)=∫Ω1k(x,t)dx,m∗(t)=∫Ωm(x,t)k(x,t)dx,c∗(t)=∫Ωc(x,t)k(x,t)dx. |
Theorem 1.1. Suppose that ∫T0m∗(t)dt>0. Then Eq (1.1) admits a unique positive periodic solution θμ(x,t) for all μ>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 λ(μ;m)<0. In the case ∫T0m∗(t)dt>0, we obtain that λ(μ;m)<0.
Next, we study the asymptotic behavior of positive periodic solutions when the diffusion rate is large.
Theorem 1.2. Suppose that
m∗(t)>0 for t∈[0,T]. | (1.3) |
Let θμ(x,t) be the unique positive periodic solution of Eq (1.1) for μ>0. Then we have
limμ→∞θμ(x,t)=ω(t) in C1,12(ˉΩ×[0,T]), | (1.4) |
where ω(t) is the unique positive periodic solution of
{ut=m∗(t)k∗(t)u−c∗(t)k∗(t)up,t∈R,u(0)=u(T). | (1.5) |
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
limμ→0+θμ(x,t)=θ(x,t) locally uniformly in Ω×[0,T], |
provided maxΩ∫T0m(x,t)dt>0. It also shows that when m∗(t)<0<∫T0maxΩm(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 μ→∞ in Section 4.
In this section, we consider the principal eigenvalue of Eq (1.2). To this end, we first study the linear initial value problem
{ut−μk(x,t)Δu−a(x,t)u=0, in Ω×(τ,T],∂u∂ν=0, on ∂Ω×(τ,T],u(x,τ)=u0(x), in Ω, | (2.1) |
where 0≤τ<T, u0∈W2,pν(Ω)(N<p<∞) and a(x,t)∈Cα,α2(ˉΩ×[τ,∞)). It is well known that there is a one-to-one correspondence between Eq (2.1) and the evolution operator Uμ(t,τ). Then we can define that u(x,t)=Uμ(t,τ)u0 is the solution of Eq (2.1). For simplicity, let X=Lp(Ω)(N<p<∞), X1=W2,pν(Ω) and
F={u∈Cα,α2(ˉΩ×R):u(⋅,t+T)=u(⋅,t) in R}. |
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 u0∈X is positive, then Uμ(t,τ)u0>0 in C1ν(ˉΩ) for 0≤τ<t≤T.
Proof. Note that X1 is compactly embedded in X. The operator Uμ(t,τ)/X1:X1→X1 can be continuously extended to the positive operator Uμ(t,τ)∈L(X,X1). Thus Uμ(t,τ)u0≥0 in X1. Since s↦Uμ(s,τ)u0 is continuous from [τ,T] to X1 and Uμ(τ,τ)u0=u0≠0, we can get that Uμ(s,τ)u0>0 in X1 as s>τ goes to τ. In addition, we have
Uμ(t,τ)u0=Uμ(t,s)Uμ(s,τ)u0, |
for τ<s<t. Thus it can be obtained that Uμ(t,τ)u0>0 for 0≤τ<t≤T.
We now study the periodic-parabolic eigenvalue problem
{ut−μk(x,t)Δu−a(x,t)u=λ(μ;a)u, in Ω×(0,T],∂u∂ν=0, on ∂Ω×(0,T],u(x,0)=u(x,T), in Ω. | (2.2) |
If there is a nontrivial solution u(x,t) of Eq (2.2), then λ(μ;a) is called the eigenvalue. In particular, if u(x,t) is positive, then λ(μ;a) is the principal eigenvalue.
Theorem 2.1. Let Kμ:=Uμ(T,0) and r be the spectral radius of Kμ. Then r is the principal eigenvalue of Kμ with positive eigenfunction u0 if and only if λ(μ;a)=−1Tlnr is the principal eigenvalue of Eq (2.2) with positive eigenfunction u(x,t)=eλ(μ;a)tUμ(t,0)u0.
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 Kμ with positive eigenfunction u0∈X1. Let u(x,t)=eλ(μ;a)tUμ(t,0)u0. Then u(x,t) satisfies
{ut−μk(x,t)Δu−a(x,t)u=λ(μ;a)u, in Ω×(0,T],∂u∂ν=0, on ∂Ω×(0,T],u(x,0)=u0=1rKμu0=eλ(μ;a)TKμu0=u(x,T), in Ω. |
According to the regularity results, we have u(x,t)∈C2+α,1+α2(ˉΩ×R). This means that μ=−1Tlnr is the principal eigenvalue of Eq (2.2), while u(x,t)=eλ(μ;a)tUμ(t,0)u0 is the corresponding positive eigenfunction.
On the contrary, suppose that λ(μ;a)=−1Tlnr is the eigenvalue of Eq (2.2) with positive eigenfunction u(x,t). Set v(x,t)=e−λ(μ;a)tu(x,t). Then v(x,t) is the solution of
{vt−μk(x,t)Δv−a(x,t)v=0, in Ω×(0,T],∂v∂ν=0, on ∂Ω×(0,T],v(x,0)=u(x)=:u0, in Ω. |
Thus, we obtain v(x,t)=Uμ(t,0)u0 for 0≤t≤T and u0∈X1 is positive. Hence,
v(T)=e−λ(μ;a)Tu0=Kμu0. |
It follows from Krein-Rutman theorem that e−λ(μ;a)T=r.
Remark 2.1. For τ<t, it follows that Uμ(t,τ) is a compact and strongly positive operator on X1. Moreover, by Krein-Rutman theorem, we obtain r>0, and r is the unique principal eigenvalue of Kμ. This implies that Eq (2.2) has the unique principal eigenvalue λ(μ;a) for any μ>0.
Lemma 2.2. Let a1(x,t), a(x,t)∈F satisfy
a1(x,t)<a2(x,t) in ˉΩ×[0,T]. |
Then λ(μ;a2)<λ(μ;a1) for any μ>0.
Proof. Assume that there exists μ1>0 such that λ(μ1;a2)≥λ(μ1;a1). Let u1(x,t) and u2(x,t) be corresponding positive eigenfunctions, chosen in such a way that
0<u1(x,t)<u2(x,t) in ˉΩ×[0,T]. |
Then ω(x,t)=u2(x,t)−u1(x,t) satisfies
{ωt−μ1k(x,t)Δω−a1(x,t)ω>λ(μ1;a1)ω, in Ω×(0,T],∂ω∂ν=0, on ∂Ω×(0,T],ω(x,0)=ω(x,T), in Ω. |
Set ϕ(x,t)=e−λ(μ1;a1)tω(x,t), then we have
{ϕt−μ1k(x,t)Δϕ−a1(x,t)ϕ>0, in Ω×(0,T],∂ϕ∂ν=0, on ∂Ω×(0,T],ϕ(x,0)=ω(x,0)=ω(x,T), in Ω. |
Thus, for any x∈Ω, we can obtain
ϕ(x,T)>Kμ1ω(x,0) and ϕ(x,T)=e−λ(μ1;a1)Tω(x,0). |
Hence, we obtain
(e−λ(μ1;a1)T−Kμ1)ω(x,0)>0 in X1. |
Note that ω(x,0)>0, it follows from [2, Theorem 7.3] that
e−λ(μ1;a1)T=rμ1<e−λ(μ1;a1)T, |
where rμ1 is the principal eigenvalue of Kμ1. This is a contradiction.
Lemma 2.3. Suppose that for any n∈N, an(x,t)∈F satisfies
limn→∞an(x,t)=a(x,t) in C1(ˉΩ×[0,T]). |
Then for fixed μ>0, we have
limn→∞λ(μ;an)=λ(μ;a). |
Proof. For any given ε>0, there exists nε∈N such that for any n>nε, there holds
a(x,t)−ε<an(x,t)<a(x,t)+ε in ˉΩ×[0,T]. |
Notice that λ(μ;a±ε)=λ(μ;a)∓ε. From Lemma 2.2, we have
λ(μ;a)−ε<λ(μ;an)<λ(μ;a)+ε, |
for any n>nε.
Lemma 2.4. Let λ(μ;a) be the principal eigenvalue of Eq (2.2) for μ>0. Then we have
λ(μ;a)≤−∫T0a∗(t)dt∫T0k∗(t)dt, | (2.3) |
here a∗(t)=∫Ωa(x,t)k(x,t)dx.
Proof. First, we consider the case
∫T0∫Ωkt(x,t)k2(x,t)dxdt≠0. |
Let φ(x,t) be the positive eigenfunction corresponding to the principal eigenvalue λ(μ;a). Taking α>0 satisfies
lnα=−∫T0∫Ωkt(x,t)lnφ(x,t)k2(x,t)dxdt∫T0∫Ωkt(x,t)k2(x,t)dxdt. |
Then φα:=αφ(x,t) is also the principal eigenfunction of Eq (2.2). It is easy to obtain
λ(μ;a)∫T0∫Ω1k(x,t)dxdt=−∫T0∫Ωa(x,t)k(x,t)dxdt−μ∫T0∫ΩΔφαφαdxdt=−∫T0∫Ωa(x,t)k(x,t)dxdt−μ∫T0∫Ω|Dφα|2φ2αdxdt. | (2.4) |
This implies that Eq (3.2) holds.
Next, we consider the case of
∫T0∫Ωkt(x,t)k2(x,t)dxdt=0. |
We can find smooth T-periodic functions {kn(x,t)} such that
limn→∞kn(x,t)=k(x,t) in C(ˉΩ×[0,T]), |
and
∫T0∫Ω(kn(x,t))tk2n(x,t)dxdt≠0. |
It follows from Lemma 2.3 that
limn→∞λn(μ;a)=λ(μ;a), |
where λn(μ;a) is the principal eigenvalue of Eq (2.2) with k(x,t) replaced by kn(x,t). It is clear from Eq (2.4) that
λn(μ;a)=−∫T0∫Ωa(x,t)kn(x,t)dxdt−μ∫T0∫Ω|Dφα|2φ2αdxdt. |
Letting n→∞, 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 λD be the principal eigenvalue of the eigenvalue problem
{ut−μk(x,t)Δu−a(x,t)u=λDu, in Ω×(0,T],u=0, on ∂Ω×(0,T],u(x,0)=u(x,T), in Ω. |
By a similar way as in [2], we can show
λD≤−1T∫T0[μk(x,s)+a(s)]ds, |
for any μ>0.
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
QT=Ω×(0,T],Q1=∂Ω×(0,T]. |
Definition 3.1. The continuous function ˉu(x,t) is called the upper-solution of Eq (1.1), if
{ut≥μk(x,t)Δu+m(x,t)u−c(x,t)up,inQT,∂u∂ν≥0,onQ1,u(x,0)≥u(x,T),inΩ, |
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 ˉu(x,t), 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 θμ(x,t)∈C1+α,(1+α)/2(ˉQT) that satisfies
u_(x,t)≤θμ(x,t)≤ˉu(x,t) in ˉQT. |
Proof. Let
f(x,t,u)=m(x,t)u−c(x,t)up. |
Then there exists a constant K>0 such that
|f(x,t,u2)−f(x,t,u1)|≤K|u2−u1|, |
for any (x,t,ui)∈ˉQT×[u_(x,t),ˉu(x,t)], i=1,2. It follows from Lp theory that for any u∈Cα,α2(ˉQT) satisfying [u_(x,t),ˉu(x,t)], the linear initial value problem
{vt−μk(x,t)Δv+Kv=Ku+f(x,t,u), in QT,∂v∂ν=0, on Q1,v(x,0)=u(x,T), in Ω, |
admits a unique solution v. Thus, a nonlinear operator v=Fu is defined. We will prove that there is a fixed point for F in four steps.
Step1. In this step, we prove that if u_≤u1≤u2≤ˉu, then u_≤v1=Fu1≤v2=Fu2≤ˉu.
Take ω1=v2−v1, then ω1 satisfies
{[ω1]t−μk(x,t)Δω1+Kω1=K(u2−u1)+f(x,t,u2)−f(x,t,u1)≥0, in QT,∂ω1∂ν=0, on Q1,ω1(x,0)=u2(x,T)−u1(x,T)≥0, in Ω. |
By the comparison principle, we obtain ω1≥0. This implies Fu2≥Fu1. Similarly, let ω2=v1−u_, then ω2 satisfies
{[ω2]t−μk(x,t)Δω2+Kω2=K(u1−u_)+f(x,t,u1)−f(x,t,u_)≥0, in QT,∂ω2∂ν=0, on Q1,ω1(x,0)=u1(x,T)−u_(x,T)≥0, in Ω. |
Thus, u_≤v1. Similarly, v2≤ˉu.
Step2. In this step, we construct a convergent monotone sequence.
The iterative sequences {un} and {vn} are constructed as follows:
u1=Fˉu,u2=Fu1,⋯,un=Fun−1⋯, |
v1=Fu_,v2=Fv1,⋯,vn=Fvn−1⋯. |
Since u_≤ˉu and F is monotonically non-decreasing, then
u_≤v1≤u1≤ˉu. |
Similarly, we obtain
u_≤vn≤un≤ˉu. |
And since u_≤u1≤ˉu,
u_≤u2≤u1≤ˉu. |
By induction, we have un+1≤un. In the same way, we obtain vn≤vn+1. Thus, we have
u_≤v1≤v2≤⋯≤u2≤u1≤ˉu. |
This also implies that {un} and {vn} are monotonically bounded sequences, so there are u0(x,t) and v0(x,t) such that
limn→∞un(x,t)=u0(x,t),limn→∞vn(x,t)=v0(x,t). |
Thus
u_(x,t)≤v0(x,t)≤u0(x,t)≤ˉu(x,t) in ˉQT. |
Step3. In this step, we prove that u0(x,t) and v0(x,t) are solutions of Eq (1.1).
Take E=W2,1p(QT)(p>n+2). First, we prove that F:D→C(ˉQT) is a compact operator, where
D={u(x,t)∈E:u_(x,t)≤u(x,t)≤ˉu(x,t) in ˉQT}. |
For u1, u2∈E, let v1=Fu1 and v2=Fu2, then ω3=v2−v1 satisfies
{[ω3]t−μk(x,t)Δω3+Kω3=K(u2−u1)+f(x,t,u2)−f(x,t,u1), in QT,∂ω3∂ν=0, on Q1,ω1(x,0)=u2(x,T)−u1(x,T), in Ω. |
By the Lp estimate and embedding theorem, it follows that
‖ω3‖C1+α,1+α2(ˉQT)≤C‖ω3‖W2,1p(QT)≤C1(‖u2−u1‖Lp(QT)+‖u2(x,T)−u1(x,T)‖Lp(Ω)), |
here C and C1 are positive constants. Thus F:D→C(ˉQT) is continuous. It is known from the embedding theorem that if u is bounded in W2,1p(QT), then Fu is bounded in C1+α,(1+α)/2(ˉQT). This means that F:D→C(ˉQT) is a compact operator.
Since un is bounded, un=Fun−1 has a convergent subsequence in C(ˉQT). By the monotonicity of un,
limn→∞un(x,t)=u0(x,t) in C(ˉQT). |
Thus u0(x,t) is the solution of Eq (1.1) in W2,1p(QT). The embedding theorem is used again to get u0(x,t)∈C1+α,(1+α)/2(ˉQT). In the same way, we get that v0(x,t) is the classical solution of Eq (1.1).
Step4. 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 τ(x,t)=u0(x,t+T)−u0(x,t) satisfies
{τt(x,t)−μk(x,t)Δτ(x,t)=m(x,t)τ(x,t)−pc(x,t)˜up−1(x,t)τ(x,t), in QT,∂τ∂ν=0, on Q1,τ(x,0)=0, in Ω, | (3.1) |
here ˜u(x,t) is between u0(x,t+T) and u0(x,t). It is well known that the solution of Eq (3.1) is unique, thus u0(x,t+T)≡u0(x,t) in ˉQT.
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 v1 and v2 are two positive periodic solutions of Eq (1.1). We first prove that there exists a large constant M>1 such that
M−1v1<v2<Mv1 in QT. |
Indeed, it is clear that there exists M1>1 such that
v2(x,0)=v2(x,T)<M1v1(x,T)=M1v1(x,0) in Ω. |
This implies v2(x,0)≢ on \bar\Omega . Let \eta(x, t): = M_1v_1(x, t)-v_2(x, t) , then \eta(x, t) satisfies
\begin{align*} \eta_t-\mu k(x,t)\Delta\eta = &m(x,t)\eta-c(x,t)[M_1v_1^p-v_2^p]\\ > &m(x,t)\eta-c(x,t)[(M_1v_1)^p-v_2^p]\\ = &m(x,t)\eta-pc(x,t)\varsigma^{p-1}(x,t)\eta, \end{align*} |
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
\begin{equation*} M^{-1}v_1 < v_2 < Mv_1\,\,\text{ in }\,\,Q_T. \end{equation*} |
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
\begin{equation*} \alpha_{*} = \inf\left\{\alpha > 0\,|\,u^*(x,t)\leq \alpha u_*(x,t)\,\text{ in } \bar Q_T \right\}. \end{equation*} |
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
\begin{equation*} \pi(x,0) = \pi(x,T)\geq\alpha_1 u_*(x,T) = \alpha_1 u_*(x,0)\,\,\text{ on }\,\,\bar\Omega, \end{equation*} |
for some small \alpha_1 > 0 . We can use the previous method to prove the existence of M to show that
\begin{equation*} \pi(x,t)\geq\alpha_1 u_*(x,t)\,\,\text{ on }\,\,\bar Q_T. \end{equation*} |
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
\begin{equation*} 0 < \varepsilon\leq\left[\frac{-\lambda(\mu;m)}{\max_{\bar Q_T}c(x,t)}\right]^{1-p}. \end{equation*} |
Take
\begin{equation*} M > \max\left\{1,\left[\frac{-\lambda(\mu;m)}{\min_{\bar Q_T}c(x,t)}\right]^{1-p}\right\}. \end{equation*} |
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
\begin{equation*} \begin{cases} u_t-\mu k(x,t)\Delta u-\left[m(x,t)-pc(x,t)\theta_\mu^{p-1}\right]u = \lambda_1u,&\text{ in } \Omega\times(0,T],\\ \frac{\partial u}{\partial\nu} = 0,&\text{ on } \partial\Omega\times(0,T],\\ u(x,0) = u(x,T),&\text{ in } \Omega. \end{cases} \end{equation*} |
Due to
\begin{equation*} m(x,t)-c(x,t)\hat{u}^{p-1} > m(x,t)-pc(x,t)\hat{u}^{p-1}, \end{equation*} |
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
\begin{equation*} (I-K_\mu)\theta_\mu(x,0) = -\int_0^TU_\mu(T,\tau)c(x,\tau)\theta_\mu^p(x,\tau)\,d\tau\,\,\text{ in }\,\,X_1. \end{equation*} |
Notice that \theta_\mu(x, t) > 0 . We now apply [2, Theorem 7.3] to obtain
\begin{equation*} e^{-\lambda(\mu;m)T} > 1. \end{equation*} |
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
\begin{equation} \lambda(\mu;m)\leq-\frac{\int_0^T m_*(t)\,dt}{\int_0^T k_*(t)\,dt}. \end{equation} | (3.2) |
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 .
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
\begin{equation} \begin{cases} u_t = \mu k(x,t)\Delta u+m(x,t)(u+\varepsilon)-c(x,t)u^{p},&\text{ in } Q_T,\\ \frac{\partial u}{\partial\nu} = 0,&\text{ on } Q_1,\\ u(x,0) = u(x,T),&\text{ in } \Omega, \end{cases} \end{equation} | (4.1) |
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
\begin{equation} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \theta_\mu(x,t)\,\,\mathit{\text{ in }}\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} | (4.2) |
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
\begin{equation*} \begin{cases} \sigma_t = k(x,\frac{t}{\mu})\Delta \sigma+\frac{1}{\mu}[m(x,\frac{t}{\mu})(\sigma+\varepsilon) -c(x,\frac{t}{\mu})\sigma^{p}],&\text{ in } Q_T,\\ \frac{\partial \sigma}{\partial\nu} = 0,&\text{ on } Q_1,\\ \sigma(x,0) = \theta^{\varepsilon}_{\mu}(x,0),&\text{ in } \Omega. \end{cases} \end{equation*} |
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
\begin{equation} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \nu(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} | (4.3) |
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
\begin{equation*} \nu(x,t) = \nu(x,0)+\mu\int_0^t[k(x,s)\Delta\nu(x,s)+m(x,s)\nu -c(x,s)\nu^p]\,ds. \end{equation*} |
It is easy to obtain
\begin{equation*} \begin{cases} \nu_t = \mu k(x,t)\Delta\nu+m(x,t)\nu-c(x,t)\nu^{p},&\text{ in } Q_T,\\ \frac{\partial\nu}{\partial\nu} = 0,&\text{ on } Q_1,\\ \nu(x,0) = \nu(x,T),&\text{ in } \Omega. \end{cases} \end{equation*} |
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
\begin{equation*} \lim\limits_{\mu\to\infty}\theta_\mu(x,t) = \theta(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T). \end{equation*} |
\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
\begin{equation*} \begin{aligned} -\int_0^T\theta_\mu(x,t) f_t(t)\,dt = &\mu \int_0^T k(x,t)f(t)\Delta\theta_\mu(x,t)\,dt\\ &+\int_0^Tm(x,t)\theta_\mu(x,t)f(t)\,dt-\int_0^T c(x,t)\theta^{p}_\mu(x,t)f(t)\,dt. \end{aligned} \end{equation*} |
By dividing \mu and making \mu\to\infty , we have
\begin{equation*} \int_0^T k(x,t)f(t)\Delta\theta(x,t)\,dt = 0. \end{equation*} |
Since f(t) is arbitrary, we obtain
\begin{equation*} \Delta\theta(x,t) = 0. \end{equation*} |
Then we derive
\begin{equation*} \int_{\Omega}|\nabla\theta(x,t)|^2\,dx = 0. \end{equation*} |
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
\begin{equation*} \int_t^{t+\varepsilon}\int_{\Omega}\frac{u_s(x,s)}{k(x,s)}\,dxds = \int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}u(x,s)\,dxds -\int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}u^p(x,s)\,dxds. \end{equation*} |
Then we have
\begin{equation*} \begin{aligned} &\int_{\Omega}\frac{u(x,t+\varepsilon)}{k(x,t+\varepsilon)}\,dx -\int_{\Omega}\frac{u(x,t)}{k(x,t)}\,dx+\int_t^{t+\varepsilon} \int_{\Omega}\frac{k_{t}(x,s)}{k^2(x,s)}u(x,s)\,dxds\\ = &\int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}u(x,s)\,dxds- \int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}u^p(x,s)\,dxds. \end{aligned} \end{equation*} |
Taking \mu\to\infty , we obtain
\begin{equation*} \begin{aligned} &\int_{\Omega}\frac{\theta(t+\varepsilon)}{k(x,t+\varepsilon)}\,dx -\int_{\Omega}\frac{\theta(t)}{k(x,t)}\,dx+\int_t^{t+\varepsilon} \int_{\Omega}\frac{k_{t}(x,s)}{k^2(x,s)}\theta(s)\,dxds\\ = &\int_t^{t+\varepsilon}\int_{\Omega}\frac{m(x,s)}{k(x,s)}\theta(s)\,dxds- \int_t^{t+\varepsilon}\int_{\Omega}\frac{c(x,s)}{k(x,s)}\theta^p(s)\,dxds. \end{aligned} \end{equation*} |
Thus, we derive
\begin{equation*} \left[\theta(t)\int_{\Omega}\frac{1}{k(x,t)}\,dx\right]_{t} = \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx\theta(t)-\int_{\Omega} \frac{c(x,t)}{k(x,t)}\,dx\theta^p(t)-\int_{\Omega}\frac{k_{t}(x,t)} {k^2(x,t)}\,dx\theta(t). \end{equation*} |
Hence,
\begin{equation*} \begin{aligned} \theta_{t}(t) = &\frac{1}{k_*(t)}\int_{\Omega} \frac{m(x,t)}{k(x,t)}\,dx\theta(t) -\frac{1}{k_*(t)}\int_{\Omega}\frac{c(x,t)}{k(x,t)}\,dx\theta^p(t)\\ &-\frac{1}{k_*(t)}\int_{\Omega}\frac{k_{t}(x,t)} {k^2(x,t)}\,dx\theta(t)-\frac{[k_*(t)]_t}{k_*(t)}\theta(t), \end{aligned} \end{equation*} |
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
\begin{equation} \int_{\Omega}\frac{u_t(x,t)}{k(x,t)}\,dx = \int_{\Omega}\frac{m(x,t)}{k(x,t)}u(x,t)\,dx -\int_{\Omega}\frac{c(x,t)}{k(x,t)}u^p(x,t)\,dx. \end{equation} | (4.4) |
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
\begin{equation*} \begin{aligned} &-\int_0^T\int_{\Omega}u(x,t)\left[\frac{f(t)}{k(x,t)}\right]_{t}\,dxdt\\ = &\int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}u(x,t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}u^p(x,t)f(t)\,dxdt. \end{aligned} \end{equation*} |
Letting \mu\to\infty , we know
\begin{equation*} -\int_0^T\int_{\Omega}\theta(t)\left[\frac{f(t)}{k(x,t)}\right]_{t}\,dxdt = \int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}\theta(t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}\theta^p(t)f(t)\,dxdt. \end{equation*} |
This implies
\begin{equation*} \int_0^T\int_{\Omega}\frac{f(t)}{k(x,t)}\theta_{t}(t)\,dxdt = \int_0^T\int_{\Omega}\frac{m(x,t)}{k(x,t)}\theta(t)f(t)\,dxdt -\int_0^T\int_{\Omega}\frac{c(x,t)}{k(x,t)}\theta^p(t)f(t)\,dxdt. \end{equation*} |
By the arbitrary of f(t) , it follows that
\begin{equation*} \int_{\Omega}\frac{1}{k(x,t)}\,dx\theta_{t}(t) = \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx\theta(t) -\int_{\Omega}\frac{c(x,t)}{k(x,t)}\,dx\theta^p(t). \end{equation*} |
Thus, we have
\begin{equation*} \begin{cases} \theta_t = \frac{\tilde{M}(t)}{\tilde{k}(t)}\theta -\frac{\tilde{C}(t)}{\tilde{k}(t)}\theta^p,\quad t\in\mathbb{R},\\ \theta(0) = \theta(T). \end{cases} \end{equation*} |
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
\begin{equation} \lim\limits_{\mu\to\infty}\theta^{\varepsilon}_\mu(x,t) = \omega^{\varepsilon}(t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation} | (4.5) |
where \omega^{\varepsilon}(t) satisfies
\begin{equation} \begin{cases} \omega^{\varepsilon}_t = \frac{M_*(t)}{k_*(t)} (\omega^{\varepsilon}+\varepsilon) -\frac{c_*(t)}{k_*(t)}(\omega^{\varepsilon})^p,\quad t\in\mathbb{R},\\ \omega^{\varepsilon}(0) = \omega^{\varepsilon}(T). \end{cases} \end{equation} | (4.6) |
Since
\begin{equation*} \int_{\Omega}\frac{m(x,t)}{k(x,t)}\,dx > 0\,\,\text{ in }\,\,[0,T], \end{equation*} |
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
\begin{equation*} \lim\limits_{\varepsilon\to0+}\omega^{\varepsilon}(t) = \omega_{0}(t)\,\,\text{ in }\,\,[0,T]. \end{equation*} |
The uniqueness of the positive solution of Eq (4.6) implies that
\begin{equation*} \lim\limits_{\varepsilon\to0+}\omega^{\varepsilon}(t) = \omega(t)\,\,\text{ in }\,\,[0,T]. \end{equation*} |
It follows from Lemma 4.1 that
\begin{equation*} \lim\limits_{\varepsilon\to0+}\theta_\mu^{\varepsilon}(x,t) = \theta_\mu(x,t)\,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T). \end{equation*} |
This means that \theta(t) is positive, together with (4.4)–(4.6). Thus, we must have
\begin{equation*} \theta(t)\equiv\omega(t)\,\,\text{ in }\,\,[0,T]. \end{equation*} |
This also implies that
\begin{equation*} \lim\limits_{\mu\to\infty}\theta_\mu(x,t) = \omega(t) \,\,\text{ in }\,\,C^{1,\frac{1}{2}}(\bar Q_T), \end{equation*} |
holds for the entire sequence.
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.
M. Fan was responsible for writing the original draft. J. Sun handled the review and supervision.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
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).
The authors declare there is no conflict of interest.
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