The novel coronavirus disease (COVID-19) pandemic has profoundly impacted the global economy and human health. The paper mainly proposed an improved susceptible-exposed-infected-recovered (SEIR) epidemic model with media coverage and limited medical resources to investigate the spread of COVID-19. We proved the positivity and boundedness of the solution. The existence and local asymptotically stability of equilibria were studied and a sufficient criterion was established for backward bifurcation. Further, we applied the proposed model to study the trend of COVID-19 in Shanghai, China, from March to April 2022. The results showed sensitivity analysis, bifurcation, and the effects of critical parameters in the COVID-19 model.
Citation: Tao Chen, Zhiming Li, Ge Zhang. Analysis of a COVID-19 model with media coverage and limited resources[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5283-5307. doi: 10.3934/mbe.2024233
[1] | Chang-Jian Wang, Jia-Yue Zhu . Analysis of global dynamics in an attraction-repulsion model with nonlinear indirect signal and logistic source. Communications in Analysis and Mechanics, 2024, 16(4): 813-835. doi: 10.3934/cam.2024035 |
[2] | Min Jiang, Dandan Liu, Rengang Huang . Boundedness and stabilization in a quasilinear chemotaxis model with nonlocal growth term and indirect signal production. Communications in Analysis and Mechanics, 2025, 17(2): 387-412. doi: 10.3934/cam.2025016 |
[3] | Yang Liu . Global attractors for a nonlinear plate equation modeling the oscillations of suspension bridges. Communications in Analysis and Mechanics, 2023, 15(3): 436-456. doi: 10.3934/cam.2023021 |
[4] | Jiangbo Han, Runzhang Xu, Chao Yang . Continuous dependence on initial data and high energy blowup time estimate for porous elastic system. Communications in Analysis and Mechanics, 2023, 15(2): 214-244. doi: 10.3934/cam.2023012 |
[5] | Yuxuan Chen . Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity. Communications in Analysis and Mechanics, 2023, 15(4): 658-694. doi: 10.3934/cam.2023033 |
[6] | Xiaotian Hao, Lingzhong Zeng . Eigenvalues of the bi-Xin-Laplacian on complete Riemannian manifolds. Communications in Analysis and Mechanics, 2023, 15(2): 162-176. doi: 10.3934/cam.2023009 |
[7] | Senhao Duan, Yue MA, Weidong Zhang . Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields. Communications in Analysis and Mechanics, 2023, 15(2): 111-131. doi: 10.3934/cam.2023007 |
[8] | Yue Pang, Xiaotong Qiu, Runzhang Xu, Yanbing Yang . The Cauchy problem for general nonlinear wave equations with doubly dispersive. Communications in Analysis and Mechanics, 2024, 16(2): 416-430. doi: 10.3934/cam.2024019 |
[9] | Long Ju, Jian Zhou, Yufeng Zhang . Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures. Communications in Analysis and Mechanics, 2023, 15(2): 24-49. doi: 10.3934/cam.2023002 |
[10] | Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu . Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011 |
The novel coronavirus disease (COVID-19) pandemic has profoundly impacted the global economy and human health. The paper mainly proposed an improved susceptible-exposed-infected-recovered (SEIR) epidemic model with media coverage and limited medical resources to investigate the spread of COVID-19. We proved the positivity and boundedness of the solution. The existence and local asymptotically stability of equilibria were studied and a sufficient criterion was established for backward bifurcation. Further, we applied the proposed model to study the trend of COVID-19 in Shanghai, China, from March to April 2022. The results showed sensitivity analysis, bifurcation, and the effects of critical parameters in the COVID-19 model.
Chemotaxis is a physiological phenomenon of organisms seeking benefits and avoiding harm, which has been widely concerned in the fields of both mathematics and biology. In order to depict such phenomena, in 1970, Keller and Segel [1] established the first mathematical model (also called the Keller-Segel model). The general form of this model is described as follows
{ut=Δu−χ∇⋅(u∇v)+f(u), x∈Ω, t>0,τvt=Δv−v+g(u), x∈Ω, t>0,u(x,0)=u0(x),v(x,0)=v0(x), x∈Ω, | (1.1) |
where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary, the value of τ can be chosen by 0 or 1 and the chemotaxis sensitivity coefficient χ>0. Here, u is the density of cell or bacteria and v stands for the concentration of chemical signal secreted by cell or bacteria. The functions f(u) and g(u) are used to characterize the growth and death of cells or bacteria and production of chemical signals, respectively.
Over the past serval decades, considerable efforts have been done on the dynamical behavior (including the global existence and boundedness, the convergence as well as the existence of blow-up solutions) of the solutions to system (1.1) (see [2,3,4,5,6,7]). Let us briefly recall some contributions among them in this direction. For example, assume that f(u)=0 and g(u)=u. For τ=1, it has been shown that the classical solutions to system (1.1) always remain globally bounded when n=1 [8]. Additionally, there will be a critical mass phenomenon to system (1.1) when n=2, namely, if the initial data u0 fulfill ∫Ωu0dx<4πχ, the classical solutions are globally bounded [9]; and if ∫Ωu0dx>4πχ, the solutions will blow up in finite time [10,11]. However, when n≥3, Winkler [12,13] showed that though the initial data satisfy some smallness conditions, the solutions will blow up either in finite or infinite time. Assume that the system (1.1) involves a non-trivial logistic source and g(u)=u. For τ=0 and f(u)≤a−μu2 with a≥0 and μ>0, Tello and Winkler [14] obtained that there exists a unique global classical solution for system (1.1) provided that n≤2,μ>0 or n≥3 and suitably large μ>0. Furthermore, for τ>0 and n≥1, suppose that Ω is a bounded convex domain. Winkler [15] proved that the system (1.1) has global classical solutions under the restriction that μ>0 is sufficiently large. When τ=1 and f(u)=u−μu2, Winkler [16] showed that nontrivial spatially homogeneous equilibrium (1μ,1μ) is globally asymptotically stable provided that the ratio μχ is sufficiently large and Ω is a convex domain. Later, based on maximal Sobolev regularity, Cao [17] also obtained the similar convergence results by removing the restrictions τ=1 and the convexity of Ω required in [16]. In addition, for the more related works in this direction, we mention that some variants of system (1.1), such as the attraction-repulsion systems (see [18,19,20,21]), the chemotaxis-haptotaxis models (see [22,23,24]), the Keller-Segel-Navier-Stokes systems (see [25,26,27,28,29,30]) and the pursuit-evasion models (see [31,32,33]), have been deeply investigated.
Recently, the Keller-Segel model with nonlinear production mechanism of the signal (i.e. g(u) is a nonlinear function with respect to u) has attracted widespread attention from scholars. For instance, when the second equation in (1.1) satisfies vt=Δv−v+g(u) with 0≤g(u)≤Kuα for K,α>0, Liu and Tao [34] obtained the global existence of classical solutions under the condition that 0<α<2n. When f(u)≤u(a−bus) and the second equation becomes 0=Δv−v+uk with k,s>0, Wang and Xiang [35] showed that if either s>k or s=k with kn−2knχ<b, the system (1.2) has global classical solutions. When the second equation in (1.1) turns into 0=Δv−1|Ω|∫Ωg(u)+g(u) for g(u)=uκ with κ>0, Winkler [36] showed that the system has a critical exponent 2n such that if κ>2n, the solution blows up in finite time; conversely, if κ<2n, the solution is globally bounded with respect to t. More results on Keller-Segel model with logistic source can be found in [6,37,38,39,40].
In addition, previous contributions also imply that diffusion functions may lead to colorful dynamic behaviors. The corresponding model can be given by
{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v), x∈Ω, t>0,vt=Δv−v+u, x∈Ω, t>0, | (1.2) |
where D(u) and S(u) are positive functions that are used to characterize the strength of diffusion and chemoattractants, respectively. When D(u) and S(u) are nonlinear functions of u, Tao-Winkler [41] and Winkler [42] proved that the existence of global classical solutions or blow-up solutions depend on the value of S(u)D(u). Namely, if S(u)D(u)≥cuα with α>2n,n≥2 and c>0 for all u>1, then for any M>0 there exist solutions that blow up in either finite or infinite time with mass ∫Ωu0=M in [42]. Later, Tao and Winkler [41] showed that such a result is optimal, i.e., if S(u)D(u)≤cuα with α<2n,n≥1 and c>0 for all u>1, then the system (1.2) possesses global classical solutions, which are bounded in Ω×(0,∞). Furthermore, Zheng [43] studied a logistic-type parabolic-elliptic system with ut=∇⋅((u+1)m−1∇u)−χ∇(u(u+1)q−1∇v)+au−bur and 0=Δv−v+u for m≥1,r>1,a≥0,b,q,χ>0. It is shown that when q+1<max{r,m+2n}, or b>b0=n(r−m)−2(r−m)n+2(r−2)χ if q+1=r, then for any sufficiently smooth initial data there exists a classical solution that is global in time and bounded. For more relevant results, please refer to [38,44,45,46].
In the Keller-Segel model mentioned above, the chemical signals are secreted by cell population, directly. Nevertheless, in reality, the production of chemical signals may go through very complex processes. For example, signal substance is not secreted directly by cell population but is produced by some other signal substance. Such a process may be described as the following system involving an indirect signal mechanism
{ut=Δu−∇⋅(u∇v)+f(u), x∈Ω, t>0,τvt=Δv−v+w,τwt=Δw−w+u, x∈Ω, t>0, | (1.3) |
where u represents the density of cell, v and w denote the concentration of chemical signal and indirect chemical signal, respectively. For τ=1, assume that f(u)=μ(u−uγ) with μ,γ>0, Zhang-Niu-Liu [47] showed that the system has global classical solutions under the condition that γ>n4+12 with n≥2. Such a boundedness result was also extended to a quasilinear system in [48,49]. Ren [50] studied system (1.3) and obtained the global existence and asymptotic behavior of generalized solutions. For τ=0, Li and Li [51] investigated the global existence and long time behavior of classical solutions for a quasilinear version of system (1.3). In [52], we extended Li and Li's results to a quasilinear system with a nonlinear indirect signal mechanism. More relevant results involving indirect signal mechanisms can be found in [53,54,55,56].
In the existing literatures, the indirect signal secretion mechanism is usually a linear function of u. However, there are very few papers that study the chemotaxis system, where chemical signal production is not only indirect but also nonlinear. Considering the complexity of biological processes, such signal production mechanisms may be more in line with the actual situation. Thus, in this paper, we study the following chemotaxis system
{ut=Δu−χ∇⋅(φ(u)∇v)−ξ∇⋅(ψ(u)∇w)+f(u), x∈Ω, t>0,0=Δv−v+vγ11, 0=Δv1−v1+uγ2, x∈Ω, t>0,0=Δw−w+wγ31, 0=Δw1−w1+uγ4, x∈Ω, t>0,∂u∂ν=∂v∂ν=∂w∂ν=∂v1∂ν=∂w1∂ν=0, x∈∂Ω, t>0, | (1.4) |
where Ω⊂Rn(n≥1) is a smoothly bounded domain and ν denotes the outward unit normal vector on ∂Ω, the parameters χ,ξ,γ2,γ4>0, and γ1,γ3≥1. The initial data u(x,0)=u0(x) satisfy some smooth conditions. Here, the nonlinear functions are assumed to satisfy
φ,ψ∈C2([0,∞)), φ(ϱ)≤ϱ(ϱ+1)θ−1 and ψ(ϱ)≤ϱ(ϱ+1)l−1 for all ϱ≥0, | (1.5) |
with θ,l∈R. The logistic source f∈C∞([0,∞)) is supposed to satisfy
f(0)≥0 and f(ϱ)≤aϱ−bϱs for all ϱ≥0, | (1.6) |
with a,b>0 and s>1. The purpose of this paper is to detect the influence of power exponents (instead of the coefficients and space dimension n) of the system (1.4) on the existence and boundedness of global classical solutions.
We state our main result as follows.
Theorem 1.1. Let Ω⊂Rn(n≥1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ3≥1. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,l∈R. If s≥max{γ1γ2+θ,γ3γ4+l}, then for any nonnegative initial data u0∈W1,∞(Ω), the system (1.4) has a nonnegative global classical solution
(u,v,v1,w,w1)∈(C0(¯Ω×[0,∞))∩C2,1(¯Ω×(0,∞)))×(C2,0(¯Ω×(0,∞)))4. |
Furthermore, this solution is bounded in Ω×(0,∞), in other words, there exists a constant C>0 such that
‖u(⋅,t)‖L∞(Ω)+‖(v(⋅,t),v1(⋅,t),w(⋅,t),w1(⋅,t))‖W1,∞(Ω)<C |
for all t>0.
The system (1.4) is a bi-attraction chemotaxis model, which can somewhat be seen as a variant of the classical attraction-repulsion system proposed by Luca [57]. In [58], Hong-Tian-Zheng studied an attraction-repulsion model with nonlinear productions and obtained the buondedness conditions which not only depend on the power exponents of the system, but also rely on the coefficients of the system as well as space dimension n. Based on [58], Zhou-Li-Zhao [59] further improved such boundedness results to some critical conditions. Compared to [58] and [59], the boundedness condition developed in Theorem 1.1 relies only on the power exponents of the system, which removes restrictions on the coefficients of the system and space dimension n. The main difficulties in the proof of Theorem 1.1 are how to reasonably deal with the integrals with power exponents in obtaining the estimate of ∫Ω(u+1)p in Lemma 3.1. Based on a prior estimates of solutions (Lemma 2.2) and some scaling techniques of inequalities, we can overcome these difficulties and then establish the conditions of global boundedness.
The rest of this paper is arranged as follows. In Sec.2, we give a result on local existence of classical solutions and get some estimates of solutions. In Sec.3, we first prove the boundedness of ∫Ω(u+1)p and then complete the proof of Theorem 1.1 based on the Moser iteration [41, Lemma A.1].
To begin with, we state a lemma involving the local existence of classical solutions and get some estimates on the solutions of system (1.4).
Lemma 2.1. Let Ω⊂Rn(n≥1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ3≥1. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,l∈R. For any nonnegative initial data u0∈W1,∞(Ω), there exists Tmax∈(0,∞] and nonnegative functions
(u,v,v1,w,w1)∈(C0(¯Ω×[0,Tmax))∩C2,1(¯Ω×(0,Tmax)))×(C2,0(¯Ω×(0,Tmax)))4, |
which solve system (1.4) in classical sense. Furthermore,
if Tmax<∞, then limt↗Tmaxsup‖u(⋅,t)‖L∞(Ω)=∞. | (2.1) |
Proof. The proof relies on the Schauder fixed point theorem and partial differential regularity theory, which is similar to [60, Lemma 2.1]. For convenience, we give a proof here. For any T∈(0,1) and the nonnegative initial data u0∈W1,∞, we set
X:=C0(¯Ω×[0,T]) and S:={u∈X|‖u(⋅,t)‖L∞(Ω)≤R for all t∈[0,T]}, |
where R:=‖u0‖L∞(Ω)+1. We can pick smooth functions φR,ψR on [0,∞) such that φR≡φ and ψR≡ψ when 0≤ϱ≤R and φR≡R and ψR≡R when ϱ≥R. It is easy to see that S is a bounded closed convex subset of X. For any ˆu∈S, let v,v1,w and w1 solve
{−Δv+v=vγ11,x∈Ω, t∈(0,T),∂v∂ν=0,x∈∂Ω, t∈(0,T), and {−Δv1+v1=ˆuγ2,x∈Ω, t∈(0,T),∂v1∂ν=0,x∈∂Ω, t∈(0,T), | (2.2) |
as well as
{−Δw+w=wγ31,x∈Ω, t∈(0,T),∂w∂ν=0,x∈∂Ω, t∈(0,T), and {−Δw1+w1=ˆuγ4,x∈Ω, t∈(0,T),∂w1∂ν=0,x∈∂Ω, t∈(0,T), | (2.3) |
respectively, in turn, let u be a solution of
{ut=Δu−χ∇⋅(φR(u)∇v)−ξ∇⋅(ψR(u)∇w)+f(u), x∈Ω, t∈(0,T),∂u∂ν=∂v∂ν=∂w∂ν=∂v1∂ν=∂w1∂ν=0, x∈∂Ω, t∈(0,T),u(x,0)=u0(x), x∈Ω. | (2.4) |
Thus, we introduce a map Φ:ˆu(∈S)↦u defined by Φ(ˆu)=u. We shall show that for any T>0 sufficiently small, Φ has a fixed point in S. Using the elliptic regularity [61, Theorem 8.34] and Morrey's theorem [62], for a certain fixed ˆu∈S, we conclude that the solutions to (2.2) satisfy v1(⋅,t)∈C1+δ(Ω) and v(⋅,t)∈C3+δ(Ω) for all δ∈(0,1), as well as the solutions to (2.3) satisfy w1(⋅,t)∈C1+δ(Ω) and w(⋅,t)∈C3+δ(Ω) for all δ∈(0,1). From the Sobolev embedding theorem and Lp−estimate, there exist mi>0,i=1,...,4 such that
‖∇v1‖L∞((0,T);Cδ(Ω))≤m1‖v1‖L∞((0,T);W2,p(Ω))≤m2‖ˆuγ2‖L∞((0,T)×Ω) |
and
‖∇w1‖L∞((0,T);Cδ(Ω))≤m3‖w1‖L∞((0,T);W2,p(Ω))≤m4‖ˆuγ4‖L∞((0,T)×Ω) |
for p>max{1,nγ1γ2,nγ3γ4}. Furthermore, we can also find mi>0,i=5,...,10 such that
‖∇v‖L∞((0,T);Cδ(Ω))≤m5‖v‖L∞((0,T);W2,p(Ω))≤m6‖vγ11‖L∞((0,T)×Ω)≤m7‖ˆuγ1γ2‖L∞((0,T)×Ω) |
and
‖∇w‖L∞((0,T);Cδ(Ω))≤m8‖w‖L∞((0,T);W2,p(Ω))≤m9‖wγ31‖L∞((0,T)×Ω)≤m10‖ˆuγ3γ4‖L∞((0,T)×Ω) |
for p>max{1,nγ1γ2,nγ3γ4}. Since ∇v,∇w∈L∞((0,T)×Ω) and u0∈Cδ(¯Ω) for all δ∈(0,1) due to the Sobolev embedding W1,∞(Ω)↪Cδ(Ω), we can infer from [63, Theorem V1.1] that u∈Cδ,δ2(¯Ω×[0,T]) and
‖u‖Cδ,δ2(¯Ω×[0,T])≤m11 for all δ∈(0,1), | (2.5) |
with some m11>0 depending only on ‖∇v‖L∞((0,T);Cδ(Ω)),‖∇w‖L∞((0,T);Cδ(Ω)) and R. Thus, we have
‖u(⋅,t)‖L∞(Ω)≤‖u0‖L∞(Ω)+‖u(⋅,t)−u0‖L∞(Ω)≤‖u0‖L∞(Ω)+m11tδ2. | (2.6) |
Hence if T<(1m11)2δ, we can obtain
‖u(⋅,t)‖L∞(Ω)≤‖u0‖L∞(Ω)+1 | (2.7) |
for all t∈[0,T], which implies that u∈S. Thus, we derive that Φ maps S into itself. We deduce that Φ is continuous. Moreover, we get from (2.5) that Φ is a compact map. Hence, by the Schauder fixed point theorem there exists a fixed point u∈S such that Φ(u)=u.
Applying the regularity theory of elliptic equations, we derive that v1(⋅,t)∈C2+δ(Ω), v(⋅,t)∈C4+δ(Ω) and w1(⋅,t)∈C2+δ(Ω), w(⋅,t)∈C4+δ(Ω) for all δ∈(0,1). Recalling (2.5), we get v1(x,t)∈C2+δ,δ2(Ω×[ι,T]), v(x,t)∈C4+δ,δ2(Ω×[ι,T]) and w1(x,t)∈C2+δ,δ2(Ω×[ι,T]), w(x,t)∈C4+δ,δ2(Ω×[ι,T]) for all δ∈(0,1) and ι∈(0,T). We use the regularity theory of parabolic equation [63, Theorem V6.1] to get
u(x,t)∈C2+δ,1+δ2(¯Ω×[ι,T]) |
for all ι∈(0,T). The solution may be prolonged in the interval [0,Tmax) with either Tmax=∞ or Tmax<∞, where in the later case
‖u(⋅.t)‖L∞(Ω)→∞ as t→Tmax. |
Additionally, since f(0)≥0, we thus get from the parabolic comparison principle that u is nonnegative. By employing the elliptic comparison principle to the second, the third, the fourth and the fifth equations in (1.4), we conclude that v,v1,w,w1 are also nonnegative. Thus, we complete the proof of Lemma 2.1.
Lemma 2.2. Let Ω⊂Rn(n≥1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ3≥1. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,l∈R. For any η1,η2,η3,η4>0 and τ>1, we can find c1,c2,c3,c4>0 which depend only on γ1,γ2,γ3,γ4,η1,η2,η3,η4,τ, such that
∫Ωwτ1≤η2∫Ω(u+1)γ4τ+c1 and ∫Ωwτ≤η1η2∫Ω(u+1)γ3γ4τ+c2, | (2.8) |
as well as
∫Ωvτ1≤η4∫Ω(u+1)γ2τ+c3 and ∫Ωvτ≤η3η4∫Ω(u+1)γ1γ2τ+c4 | (2.9) |
for all t∈(0,Tmax).
Proof. Integrating the first equation of system (1.4) over Ω and using Hölder's inequality, it is easy to get that
ddt∫Ωudx≤∫Ωau−bus≤a∫Ωu−b|Ω|s−1(∫Ωu)s for all t∈(0,Tmax). | (2.10) |
Employing the standard ODE comparison theory, we conclude
∫Ωu≤max{∫Ωu0,(ab)1s−1|Ω|} for all t∈(0,Tmax). | (2.11) |
Moreover, integrating the fifth equation of system (1.4) over Ω, one may get
‖w1‖L1(Ω)=‖uγ4‖L1(Ω)≤‖(u+1)γ4‖L1(Ω) for all t∈(0,Tmax). | (2.12) |
For any τ>1, multiplying the fifth equation of system(1.4) with wτ−11, we can get by integration by parts that
4(τ−1)τ2∫Ω|∇wτ21|2+∫Ωwτ1=∫Ωuγ4wτ−11≤τ−1τ∫Ωwτ1+1τ∫Ωuγ4τ, | (2.13) |
where Young's inequality has been used. Thus, we deduce
‖w1‖Lτ(Ω)≤‖uγ4‖Lτ(Ω)≤‖(u+1)γ4‖Lτ(Ω) for all t∈(0,Tmax), | (2.14) |
and
4(τ−1)τ∫Ω|∇wτ21|2≤∫Ωuγ4τ≤∫Ω(u+1)γ4τ for all t∈(0,Tmax). | (2.15) |
Using Ehrling's lemma, we know that for any η2>0 and τ>1, there exists c5=c5(η2,τ)>0 such that
‖ϕ‖2L2(Ω)≤η2‖ϕ‖2W1,2(Ω)+c5‖ϕ‖2L2τ(Ω) for all ϕ∈W1,2(Ω). | (2.16) |
Let ϕ=wτ21. Combining (2.12) with (2.14), (2.15), there exists c6=c6(η2,γ4,τ)>0 such that
∫Ωwτ1≤η2∫Ω(u+1)γ4τ+c6‖(u+1)γ4‖τL1(Ω). | (2.17) |
If γ4∈(0,1], by (2.11) and Hölder's inequality, we can derive
‖(u+1)γ4‖τL1(Ω)≤c7, | (2.18) |
with c7=c7(η2,τ,γ4)>0. If γ4∈(1,∞), invoking interpolation inequality and Young's inequality, we can get from (2.11) that
‖(u+1)γ4‖τL1(Ω)≤‖(u+1)γ4‖τρLτ(Ω)‖(u+1)γ4‖τ(1−ρ)L1γ4(Ω)≤η2∫Ω(u+1)γ4τ+c8, | (2.19) |
where ρ=γ4−1γ4−1τ∈(0,1) and c8=c8(η2,τ,γ4)>0. Collecting (2.17)–(2.19), we can directly infer that the first inequality of (2.8) holds. Integrating the fourth equation of system (1.4) over Ω, we have ‖w‖L1(Ω)=‖wγ31‖L1(Ω) for all t∈(0,Tmax). Due to γ3≥1, from the first inequality of (2.8), it is easy to see that
‖w‖L1(Ω)=∫Ωwγ31≤η2∫Ω(u+1)γ3γ4+˜c1 | (2.20) |
for all t∈(0,Tmax), where ˜c1=˜c1(η2,γ3,γ4)>0. By the same procedures as in (2.13)-(2.19), we thus can obtain for any η1>0 and τ>1 that
∫Ωwτ≤η1∫Ωwγ3τ1+c9 for all t∈(0,Tmax), | (2.21) |
where c9=c9(η1,τ,γ3)>0. Recalling γ3≥1 and using the first inequality of (2.8) again, we get that
∫Ωwγ3τ1≤η2∫Ω(u+1)γ3γ4τ+c10 for all t∈(0,Tmax), | (2.22) |
with c10>0. Hence, the second inequality of (2.8) can be obtained from (2.21) and (2.22). In addition, we can employ the same processes as above to prove (2.9). Here, we omit the detailed proof. Thus, the proof of Lemma 2.2 is complete.
In order to prove the global existence and uniform boundedness of classical solutions to system (1.4), we established the following Lp−estimate for component u.
Lemma 3.1. Let Ω⊂Rn(n≥1) be a bounded domain with smooth boundary and the parameters fulfill ξ,χ,γ2,γ4>0 and γ1,γ3≥1. Assume that the nonlinear functions φ,ψ and f satisfy the conditions (1.5) and (1.6) with a,b>0,s>1 and θ,l∈R. If s≥max{γ1γ2+θ,γ3γ4+l}, for any p>max{1,1−θ,1−l,γ1γ2−s+1,γ3γ4−s+1}, there exists C>0 such that
∫Ω(u+1)p≤C | (3.1) |
for all t∈(0,Tmax).
Proof. For any p>1, we multiply the first equation of system (1.4) with (u+1)p−1 and use integration by parts over Ω to obtain
1pddt∫Ω(u+1)p≤−4(p−1)p2∫Ω|∇(u+1)p2|2+χ(p−1)∫Ω(u+1)p−2φ(u)∇u⋅∇v+ξ(p−1)∫Ω(u+1)p−2ψ(u)∇u⋅∇w+a∫Ωu(u+1)p−1−b∫Ωus(u+1)p−1 | (3.2) |
for all t∈(0,Tmax). Let Ψ1(y)=∫y0(ζ+1)p−2ψ(ζ)dζ and Ψ2(y)=∫y0(ζ+1)p−2φ(ζ)dζ. It is easy to get
∇Ψ1(u)=(u+1)p−2ψ(u)∇u | (3.3) |
and
∇Ψ2(u)=(u+1)p−2φ(u)∇u | (3.4) |
for all t∈(0,Tmax). Furthermore, by a simple calculation, one can get
|Ψ1(u)|≤1p+l−1(u+1)p+l−1 | (3.5) |
and
|Ψ2(u)|≤1p+θ−1(u+1)p+θ−1 | (3.6) |
for all t∈(0,Tmax). Thus, the second term on the right-hand side of (3.2) can be estimated as
χ(p−1)∫Ω(u+1)p−2φ(u)∇u⋅∇v=χ(p−1)∫Ω∇Ψ2(u)⋅∇v≤χ(p−1)∫ΩΨ2(u)|Δv|≤χ(p−1)p+θ−1∫Ω(u+1)p+θ−1|Δv| | (3.7) |
for all t∈(0,Tmax). Similarly, we can deduce
ξ(p−1)∫Ω(u+1)p−2ψ(u)∇u⋅∇w≤ξ(p−1)p+l−1∫Ω(u+1)p+l−1|Δw| | (3.8) |
for all t∈(0,Tmax). From the basic inequality (u+1)s≤2s(us+1) with s>0 and u≥0, we can get
−b∫Ωus(u+1)p−1≤−b2s∫Ω(u+1)p+s−1+b∫Ω(u+1)p−1 | (3.9) |
for all t∈(0,Tmax). Set m=max{a,b}. From (3.7)–(3.9), the (3.2) can be rewritten as
1pddt∫Ω(u+1)p≤χ(p−1)p+θ−1∫Ω(u+1)p+θ−1⋅|v−vγ11|+ξ(p−1)p+l−1∫Ω(u+1)p+l−1⋅|w−wγ31|+m∫Ω(u+1)p−b2s∫Ω(u+1)p+s−1≤χ(p−1)p+θ−1∫Ω(u+1)p+θ−1v+χ(p−1)p+θ−1∫Ω(u+1)p+θ−1vγ11+ξ(p−1)p+l−1∫Ω(u+1)p+l−1w+ξ(p−1)p+l−1∫Ω(u+1)p+l−1wγ31+m∫Ω(u+1)p−b2s∫Ω(u+1)p+s−1 | (3.10) |
for all t∈(0,Tmax), where we have used the equations 0=Δv−v+vγ11 and 0=Δw1−w1+uγ4 in system (1.4). In the following, we shall establish the Lp− estimate of component u.
Case (ⅰ) s>max{γ1γ2+θ,γ3γ4+l}.
It follows from Young's inequality that
∫Ω(u+1)p+θ−1vγ11≤b(p+θ−1)2s+4χ(p−1)∫Ω(u+1)p+s−1+c11∫Ωv(p+s−1)γ1s−θ1 | (3.11) |
for all t∈(0,Tmax), with c11=(2s+4χ(p−1)b(p+θ−1))p+θ−1s−θ>0. Due to s−θ>γ1γ2, we infer from Young's inequality and Lemma 2.2 by choosing τ=p+s−1γ2 that
∫Ωv(p+s−1)γ1s−θ1≤b(p+θ−1)2s+4χη4(p−1)c11∫Ωvp+s−1γ21+c12≤b(p+θ−1)2s+4χ(p−1)c11∫Ω(u+1)p+s−1+c13 | (3.12) |
for all t∈(0,Tmax), with c12=(2s+4χη4(p−1)c11b(p+θ−1))γ1γ2s−θ−γ1γ2|Ω| and c13=c12+c3. According to Young's inequality, we can find c14=(2s+4χ(p−1)b(p+θ−1))p+θ−1s−θ>0 such that
∫Ω(u+1)p+θ−1v≤b(p+θ−1)2s+4χ(p−1)∫Ω(u+1)p+s−1+c14∫Ωvp+s−1s−θ | (3.13) |
for all t∈(0,Tmax). For s−θ>γ1γ2, we use Lemma 2.2 with τ=p+s−1s−θ and Young's inequality to get
∫Ωvp+s−1s−θ≤η3η4∫Ω(u+1)γ1γ2(p+s−1)s−θ+c4≤b(p+θ−1)2s+4χ(p−1)c14∫Ω(u+1)p+s−1+c15 | (3.14) |
for all t∈(0,Tmax), with c15=(η3η4)s−θs−θ−γ1γ2⋅(2s+4χ(p−1)c14b(p+θ−1))γ1γ2s−θ−γ1γ2+c4. Analogously, we have
∫Ω(u+1)p+l−1wγ31≤b(p+l−1)2s+4ξ(p−1)∫Ω(u+1)p+s−1+c16∫Ωw(p+s−1)γ3s−l1 | (3.15) |
for all t∈(0,Tmax), where c16=(2s+4ξ(p−1)b(p+l−1))p+l−1s−l. Since s−l>γ3γ4, it follows from Young's inequality and Lemma 2.2 with τ=p+s−1γ4 that
∫Ωw(p+s−1)γ3s−l1≤b(p+l−1)2s+4ξ(p−1)c16η2∫Ωwp+s−1γ41+c17≤b(p+l−1)2s+4ξ(p−1)c16∫Ω(u+1)p+s−1+c18 | (3.16) |
for all t∈(0,Tmax), where c17=(2s+4ξ(p−1)c16η2b(p+l−1))γ3γ4s−l−γ3γ4|Ω| and c18=c17+c1. Similarly, there exists c19=(2s+4ξ(p−1)b(p+l−1))p+l−1s−l>0 such that
∫Ω(u+1)p+l−1w≤b(p+l−1)2s+4ξ(p−1)∫Ω(u+1)p+s−1+c19∫Ωwp+s−1s−l | (3.17) |
for all t∈(0,Tmax). Using Lemma 2.2 once more, one may obtain
∫Ωwp+s−1s−l≤η1η2∫Ω(u+1)γ3γ4(p+s−1)s−l+c2≤b(p+l−1)2s+4ξ(p−1)c19∫Ω(u+1)p+s−1+c20 | (3.18) |
for all t∈(0,Tmax), where c20=(η1η2)s−ls−l−γ3γ4⋅(2s+4ξ(p−1)c19b(p+l−1))γ3γ4s−l−γ3γ4+c2. Due to s>1, we thus have
∫Ω(u+1)p≤c21∫Ω(u+1)p+s−1+c22 | (3.19) |
for all t∈(0,Tmax), where c21=b2s+2(m+1) and c22=(2s+2(m+1)b)ps−1|Ω|. From (3.11)-(3.19), the inequality (3.10) can be estimated as
1pddt∫Ω(u+1)p+∫Ω(u+1)p≤χ(p−1)p+θ−1[b(p+θ−1)2s+2χ(p−1)∫Ω(u+1)p+s−1+c11c13++c14c15]+ξ(p−1)p+l−1[b(p+l−1)2s+2ξ(p−1)∫Ω(u+1)p+s−1+c16c18+c19c20]−b2s∫Ω(u+1)p+s−1+b2s+2∫Ω(u+1)p+s−1+c22(m+1)≤−b2s+2∫Ω(u+1)p+s−1+c23 | (3.20) |
for all t∈(0,Tmax), where c23=(c11c13+c14c15)⋅χ(p−1)p+θ−1+(c16c18+c19c20)⋅ξ(p−1)p+l−1+c22(m+1). Hence, we can derive (3.1) easily by using the ODE comparison principle.
Case (ⅱ) s=max{γ1γ2+θ,γ3γ4+l}.
(a) s=γ1γ2+θ=γ3γ4+l. Recalling (3.11), (3.13), (3.15) and (3.17), there hold
∫Ω(u+1)p+θ−1vγ11≤b(p+θ−1)2s+4χ(p−1)∫Ω(u+1)p+s−1+c11∫Ωv(p+s−1)γ1s−θ1 | (3.21) |
and
∫Ω(u+1)p+θ−1v≤b(p+θ−1)2s+4χ(p−1)∫Ω(u+1)p+s−1+c14∫Ωvp+s−1s−θ | (3.22) |
and
∫Ω(u+1)p+l−1wγ31≤b(p+l−1)2s+4ξ(p−1)∫Ω(u+1)p+s−1+c16∫Ωw(p+s−1)γ3s−l1 | (3.23) |
as well as
∫Ω(u+1)p+l−1w≤b(p+l−1)2s+4ξ(p−1)∫Ω(u+1)p+s−1+c19∫Ωwp+s−1s−l | (3.24) |
for all t\in (0, T_{\max}).
Since s-\theta = \gamma_{1}\gamma_{2} and s-l = \gamma_{3}\gamma_{4}. Thus, we can directly get from Lemma 2.2 that
\begin{align} \int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} = \int_{\Omega}w_{1}^\frac{p+s-1}{\gamma_{4}} \leq\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{1} \end{align} | (3.25) |
and
\begin{align} \int_{\Omega}w^\frac{p+s-1}{s-l} = \int_{\Omega}w^\frac{p+s-1}{\gamma_{3}\gamma_{4}} \leq\eta_{1}\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{2}, \end{align} | (3.26) |
and
\begin{align} \int_{\Omega}v_{1}^\frac{(p+s-1)\gamma_{1}}{s-\theta} = \int_{\Omega}v_{1}^\frac{p+s-1}{\gamma_{2}} \leq\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{3} \end{align} | (3.27) |
as well as
\begin{align} \int_{\Omega}v^\frac{p+s-1}{s-\theta} = \int_{\Omega}v^\frac{p+s-1}{\gamma_{1}\gamma_{2}} \leq\eta_{3}\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{4} \end{align} | (3.28) |
for all t\in (0, T_{\max}). For the arbitrariness of \eta_{1}, \eta_{2}, \eta_{3}, \eta_{4} > 0, we choose \eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{16}\xi(p-1)}, \eta_{1}\eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{19}\xi(p-1)}, \eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{11}\chi(p-1)} and \eta_{3}\eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{14}\chi(p-1)} in(3.25)-(3.28), respectively. Combining (3.19) with (3.21)-(3.28), the inequality (3.10) can be rewritten as
\begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+1)^{p}+\int_{\Omega}(u+1)^{p} \leq& \frac{\chi(p-1)}{p+\theta-1} \bigg[\frac{b(p+\theta-1)}{2^{s+2}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{3}c_{11}+c_{4}c_{14}\bigg] \\ &+\frac{\xi(p-1)}{p+l-1} \bigg[\frac{b(p+l-1)}{2^{s+2}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{1}c_{16}+c_{2}c_{19}\bigg] \\ &-\frac{b}{2^{s}}\int_{\Omega}(u+1)^{p+s-1}+\frac{b}{2^{s+2}} \int_{\Omega}(u+1)^{p+s-1}+c_{22}(m+1) \\ \leq& -\frac{b}{2^{s+2}}\int_{\Omega}(u+1)^{p+s-1}+c_{24} \end{align} | (3.29) |
for all t\in (0, T_{\max}), where c_{24} = \big(c_{3}c_{11}+c_{4}c_{14}\big)\cdot\frac{\chi(p-1)}{p+\theta-1}+ \big(c_{1}c_{16}+c_{2}c_{19}\big)\cdot\frac{\xi(p-1)}{p+l-1}+c_{22}(m+1). From the ODE comparison principle, we can get the desired conclusion (3.1).
(b) s = \gamma_{1}\gamma_{2}+\theta > \gamma_{3}\gamma_{4}+l. Recalling (3.11) and (3.13) again, there hold
\begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v_{1}^{\gamma_{1}} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{11}\int_{\Omega}v_{1}^{\frac{(p+s-1)\gamma_{1}}{s-\theta}} \end{align} | (3.30) |
and
\begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v \leq\frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{14}\int_{\Omega}v^{\frac{p+s-1}{s-\theta}}. \end{align} | (3.31) |
For s = \gamma_{1}\gamma_{2}+\theta, we can get from Lemma 2.2 that
\begin{align} \int_{\Omega}v_{1}^\frac{(p+s-1)\gamma_{1}}{s-\theta} = \int_{\Omega}v_{1}^\frac{p+s-1}{\gamma_{2}} \leq\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{3} \end{align} | (3.32) |
and
\begin{align} \int_{\Omega}v^\frac{p+s-1}{s-\theta} = \int_{\Omega}v^\frac{p+s-1}{\gamma_{1}\gamma_{2}} \leq\eta_{3}\eta_{4}\int_{\Omega}(u+1)^{p+s-1}+c_{4} \end{align} | (3.33) |
for all t\in (0, T_{\max}). Here, we also choose \eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{11}\chi(p-1)} in (3.32) and \eta_{3}\eta_{4} = \frac{b(p+\theta-1)}{2^{s+4}c_{14}\chi(p-1)} in (3.33). For s > \gamma_{3}\gamma_{4}+l, we can conclude from (3.15)-(3.18) that
\begin{align} \int_{\Omega}(u+1)^{p+l-1}w_{1}^{\gamma_{3}}&\leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{16}\int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} \end{align} | (3.34) |
for all t\in (0, T_{\max}), with c_{16} = \left(\frac{2^{s+4}\xi(p-1)}{b(p+l-1)}\right)^{\frac{p+l-1}{s-l}}. Moreover, using Lemma 2.2, it is easy to get
\begin{align} \int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)c_{16}\eta_{2}}\int_{\Omega} w_{1}^{\frac{p+s-1}{\gamma_{4}}}+c_{17} \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)c_{16}}\int_{\Omega}(u+1)^{p+s-1}+c_{18} \end{align} | (3.35) |
for all t\in (0, T_{\max}), where c_{17} = \left(\frac{2^{s+4}\xi(p-1)c_{16}\eta_{2}}{b(p+l-1)}\right) ^{\frac{\gamma_{3}\gamma_{4}}{s-l-\gamma_{3}\gamma_{4}}}|\Omega| and c_{18} = c_{17}+c_{1}. By a simple calculation, we know
\begin{align} \int_{\Omega}(u+1)^{p+l-1}w \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{19}\int_{\Omega}w^{\frac{p+s-1}{s-l}} \end{align} | (3.36) |
and
\begin{align} \int_{\Omega}w^{\frac{p+s-1}{s-l}}&\leq \eta_{1}\eta_{2}\int_{\Omega}(u+1)^{\frac{\gamma_{3}\gamma_{4} (p+s-1)}{s-l}}+c_{2} \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)c_{19}}\int_{\Omega}(u+1)^{p+s-1}+c_{20} \end{align} | (3.37) |
for all t\in (0, T_{\max}), where c_{19} = \left(\frac{2^{s+4}\xi(p-1)}{b(p+l-1)}\right) ^\frac{p+l-1}{s-l} and c_{20} = (\eta_{1}\eta_{2})^{\frac{s-l}{s-l-\gamma_{3}\gamma_{4}}}\cdot\left(\frac{2^{s+4}\xi(p-1) c_{19}}{b(p+l-1)}\right) ^{\frac{\gamma_{3}\gamma_{4}}{s-l-\gamma_{3}\gamma_{4}}}+c_{2}. Recalling (3.19), for s > 1, we have
\begin{align} \int_{\Omega}(u+1)^{p}\leq c_{21}\int_{\Omega}(u+1)^{p+s-1}+c_{22} \end{align} | (3.38) |
for all t\in (0, T_{\max}), where c_{21} = \frac{b}{2^{s+2}(m+1)} and c_{22} = \left(\frac{2^{s+2}(m+1)}{b}\right)^{\frac{p}{s-1}}|\Omega|. Collecting (3.30)-(3.38), it can be deduced from (3.10) that
\begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+1)^{p}+\int_{\Omega}(u+1)^{p} \leq& \frac{\chi(p-1)}{p+\theta-1} \bigg[\frac{b(p+\theta-1)}{2^{s+2}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{3}c_{11}+c_{4}c_{14}\bigg] \\ &+\frac{\xi(p-1)}{p+l-1} \bigg[\frac{b(p+l-1)}{2^{s+2}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{16}c_{18}+c_{19}c_{20}\bigg] \\ &-\frac{b}{2^{s}}\int_{\Omega}(u+1)^{p+s-1}+\frac{b}{2^{s+2}} \int_{\Omega}(u+1)^{p+s-1}+c_{22}(m+1) \\ \leq& -\frac{b}{2^{s+2}}\int_{\Omega}(u+1)^{p+s-1}+c_{25} \end{align} | (3.39) |
for all t\in (0, T_{\max}), where c_{25} = \big(c_{3}c_{11}+c_{4}c_{14}\big)\cdot\frac{\chi(p-1)}{p+\theta-1}+ \big(c_{16}c_{18}+c_{19}c_{20}\big)\cdot\frac{\xi(p-1)}{p+l-1}+c_{22}(m+1). In view of the ODE comparison principle, we conclude (3.1), directly.
(c) s = \gamma_{3}\gamma_{4}+l > \gamma_{1}\gamma_{2}+\theta. The proof of this case is similar to the case (b). Using (3.15) and (3.17) again, we get
\begin{align} \int_{\Omega}(u+1)^{p+l-1}w_{1}^{\gamma_{3}}&\leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{16}\int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} \end{align} | (3.40) |
and
\begin{align} \int_{\Omega}(u+1)^{p+l-1}w \leq \frac{b(p+l-1)}{2^{s+4}\xi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{19}\int_{\Omega}w^{\frac{p+s-1}{s-l}} \end{align} | (3.41) |
for all t\in (0, T_{\max}). Since s = \gamma_{3}\gamma_{4}+l, it is easy to deduce from Lemma 2.2 that
\begin{align} \int_{\Omega}w_{1}^\frac{(p+s-1)\gamma_{3}}{s-l} = \int_{\Omega}w_{1}^\frac{p+s-1}{\gamma_{4}} \leq\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{1} \end{align} | (3.42) |
and
\begin{align} \int_{\Omega}w^\frac{p+s-1}{s-l} = \int_{\Omega}v^\frac{p+s-1}{\gamma_{3}\gamma_{4}} \leq\eta_{1}\eta_{2}\int_{\Omega}(u+1)^{p+s-1}+c_{2} \end{align} | (3.43) |
for all t\in(0, T_{\max}). Due to the arbitrariness of \eta_{1} and \eta_{2}, here we let \eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{16}\xi(p-1)} in (3.42) and \eta_{1}\eta_{2} = \frac{b(p+l-1)}{2^{s+4}c_{19}\xi(p-1)} in (3.43). Since s > \gamma_{1}\gamma_{2}+\theta, we can derive from (3.11)-(3.14) that
\begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v_{1}^{\gamma_{1}} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{11}\int_{\Omega}v_{1}^{\frac{(p+s-1)\gamma_{1}}{s-\theta}} \end{align} | (3.44) |
for all t\in(0, T_{\max}), with c_{11} = \left(\frac{2^{s+4}\chi(p-1)}{b(p+\theta-1)}\right)^{\frac{p+\theta-1}{s-\theta}} > 0. Due to s-\theta > \gamma_{1}\gamma_{2}, from Young's inequality and Lemma 2.2, we can obtain
\begin{align} \int_{\Omega}v_{1}^{\frac{(p+s-1)\gamma_{1}}{s-\theta}} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi \eta_{4}(p-1)c_{11}}\int_{\Omega}v_{1}^{\frac{p+s-1}{\gamma_{2}}}+c_{12} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)c_{11}}\int_{\Omega}(u+1)^{p+s-1}+c_{13} \end{align} | (3.45) |
for all t\in(0, T_{\max}), with c_{12} = \left(\frac{2^{s+4}\chi \eta_{4}(p-1)c_{11}}{b(p+\theta-1)}\right) ^{\frac{\gamma_{1}\gamma_{2}}{s-\theta-\gamma_{1}\gamma_{2}}}|\Omega| and c_{13} = c_{12}+c_{3}. In view of Young's inequality, it is easy to get
\begin{align} \int_{\Omega}(u+1)^{p+\theta-1}v \leq\frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)}\int_{\Omega}(u+1)^{p+s-1} +c_{14}\int_{\Omega}v^{\frac{p+s-1}{s-\theta}} \end{align} | (3.46) |
for all t\in (0, T_{\max}), with c_{14} = \left(\frac{2^{s+4}\chi(p-1)}{b(p+\theta-1)}\right) ^{\frac{p+\theta-1}{s-\theta}} > 0. Due to s-\theta > \gamma_{1}\gamma_{2}, thus we use Lemma 2.2 with \tau = \frac{p+s-1}{s-\theta} and Young's inequality to obtain
\begin{align} \int_{\Omega}v^{\frac{p+s-1}{s-\theta}} \leq \eta_{3}\eta_{4}\int_{\Omega}(u+1) ^{\frac{\gamma_{1}\gamma_{2}(p+s-1)}{s-\theta}}+c_{4} \leq \frac{b(p+\theta-1)}{2^{s+4}\chi(p-1)c_{14}}\int_{\Omega}(u+1)^{p+s-1}+c_{15} \end{align} | (3.47) |
for all t\in (0, T_{\max}), with c_{15} = (\eta_{3}\eta_{4})^{\frac{s-\theta}{s-\theta-\gamma_{1}\gamma_{2}}}\cdot\left(\frac{2^{s+4}\chi(p-1)c_{14}}{b(p+\theta-1)}\right) ^{\frac{\gamma_{1}\gamma_{2}}{s-\theta-\gamma_{1}\gamma_{2}}}+c_{4}. For s > 1, we get from (3.19) that
\begin{align} \int_{\Omega}(u+1)^{p}\leq c_{21}\int_{\Omega}(u+1)^{p+s-1}+c_{22} \end{align} | (3.48) |
for all t\in (0, T_{\max}), where c_{21} = \frac{b}{2^{s+2}(m+1)} and c_{22} = \left(\frac{2^{s+2}(m+1)}{b}\right)^{\frac{p}{s-1}}|\Omega|. Collecting (3.40)–(3.48), we can infer from (3.10) that
\begin{align} \frac{1}{p}\frac{d}{dt}\int_{\Omega}(u+1)^{p}+\int_{\Omega}(u+1)^{p} \leq-\frac{b}{2^{s+2}}\int_{\Omega}(u+1)^{p+s-1}+c_{26} \end{align} | (3.49) |
for all t\in (0, T_{\max}), where c_{26} = \big(c_{11}c_{13}+c_{14}c_{15}\big)\cdot\frac{\chi(p-1)}{p+\theta-1}+ \big(c_{1}c_{16}+c_{2}c_{19}\big)\cdot\frac{\xi(p-1)}{p+l-1}+c_{22}(m+1). Hence, we can conclude (3.1) by using the ODE comparison principle. Thus, we complete the proof of Lemma 3.1.
Now, we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1 Let \Omega\subset\mathbb{R}^n(n\geq1) be a bounded domain with smooth boundary and the parameters fulfill \xi, \chi, \gamma_{2}, \gamma_{4} > 0 and \gamma_{1}, \gamma_{3}\geq1. Assume that the nonlinear functions \varphi, \psi and f satisfy the conditions (1.5) and (1.6) with a, b > 0, s > 1 and \theta, l\in \mathbb{R}. According to Lemma 3.1, for any p > \max\{1, 1-\theta, 1-l, n\gamma_{1}\gamma_{2}, n\gamma_{3}\gamma_{4}, \gamma_{1}\gamma_{2}-s+1, \gamma_{3}\gamma_{4}-s+1\}, there exists c_{27} > 0 such that \|u\|_{L^{p}(\Omega)}\leq c_{27} for all t\in(0, T_{\max}). We deal with the second, the third, the fourth and the fifth equations in system (1.4) by elliptic L^{p}- estimate to obtain
\begin{align} \|v(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{1}\gamma_{2}}}(\Omega)} +\|v_{1}(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{2}}}(\Omega)} +\|w(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{3}\gamma_{4}}}(\Omega)} +\|w_{1}(\cdot, t)\|_{W^{2, \frac{p}{\gamma_{4}}}(\Omega)} \leq c_{28} \end{align} | (3.50) |
for all t\in(0, T_{\max}), with some c_{28} > 0. Applying the Sobolev imbedding theorem, we can infer that
\begin{align} \|v(\cdot, t)\|_{W^{1, \infty}} +\|v_{1}(\cdot, t)\|_{W^{1, \infty}} +\|w(\cdot, t)\|_{W^{1, \infty}} +\|w_{1}(\cdot, t)\|_{W^{1, \infty}} \leq c_{29} \end{align} | (3.51) |
for all t\in(0, T_{\max}), with some c_{29} > 0. In view of Moser iteration [41, Lemma A.1], there exists c_{30} > 0 such that
\|u(\cdot, t)\|_{L^{ \infty}(\Omega)}\leq c_{30} |
for all t\in(0, T_{\max}), which combining with Lemma 2.1 implies that T_{\max} = \infty. The proof of Theorem 1.1 is complete.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
We would like to thank the anonymous referees for many useful comments and suggestions that greatly improve the work. This work was partially supported by NSFC Grant NO. 12271466, Scientific and Technological Key Projects of Henan Province NO. 232102310227, NO. 222102320425 and Nanhu Scholars Program for Young Scholars of XYNU NO. 2020017.
The authors declare there is no conflict of interest.
[1] |
K. Sarkar, J. Mondal, S. Khajanchi, How do the contaminated environment influence the transmission dynamics of COVID-19 pandemic, Eur. Phys. J. Spec. Top., 231 (2022), 3697–3716. https://doi.org/10.1140/epjs/s11734-022-00648-w doi: 10.1140/epjs/s11734-022-00648-w
![]() |
[2] | World Health Organization, Coronavirus disease (COVID-19) pandemic, 2023. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019. |
[3] |
W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
![]() |
[4] |
L. Zhou, M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), 312–324. https://doi.org/10.1016/j.nonrwa.2011.07.036 doi: 10.1016/j.nonrwa.2011.07.036
![]() |
[5] |
R. George, N. Gul, A. Zeb, Z. Avazzadeh, S. Djilali, S. Rezapour, Bifurcations analysis of a discrete time SIR epidemic model with nonlinear incidence function, Results Phys., 38 (2022), 105580. https://doi.org/10.1016/j.rinp.2022.105580 doi: 10.1016/j.rinp.2022.105580
![]() |
[6] |
R. M. Anderson, R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361–367. https://doi.org/10.1038/280361a0 doi: 10.1038/280361a0
![]() |
[7] |
Z. Hu, P. Bi, W. Ma, S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 93–112. https://doi.org/10.3934/dcdsb.2011.15.93 doi: 10.3934/dcdsb.2011.15.93
![]() |
[8] |
A. Lahrouz, L. Omari, D. Kiouach, A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Appl. Math. Comput., 218 (2012), 6519–6525. https://doi.org/10.1016/j.amc.2011.12.024 doi: 10.1016/j.amc.2011.12.024
![]() |
[9] |
S. Gao, H. Ouyang, J. J. Nieto, Mixed vaccination strategy in SIRS epidemic model with seasonal variability on infection, Int. J. Biomath., 4 (2011), 473–491. https://doi.org/10.1142/S1793524511001337 doi: 10.1142/S1793524511001337
![]() |
[10] |
J. Li, Z. Ma, Global analysis of SIS epidemic models with variable total population size, Math. Comput. Modell., 39 (2004), 1231–1242. https://doi.org/10.1016/j.mcm.2004.06.004 doi: 10.1016/j.mcm.2004.06.004
![]() |
[11] |
Y. Li, J. Cui, The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2353–2365. https://doi.org/10.1016/j.cnsns.2008.06.024 doi: 10.1016/j.cnsns.2008.06.024
![]() |
[12] |
M. E. Fatini, A. Lahrouz, R. Pettersson, A. Settati, R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput., 316 (2018), 326–341. https://doi.org/10.1016/j.amc.2017.08.037 doi: 10.1016/j.amc.2017.08.037
![]() |
[13] |
Y. Ding, Y. Fu, Y. Kang, Stochastic analysis of COVID-19 by a SEIR model with Lévy noise, Chaos: Interdiscip. J. Nonlinear Sci., 31 (2021), 043132. https://doi.org/10.1063/5.0021108 doi: 10.1063/5.0021108
![]() |
[14] |
M. Cai, G. E. Karniadakis, C. Li, Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant, Chaos: Interdiscip. J. Nonlinear Sci., 32 (2022), 071101. https://doi.org/10.1063/5.0099450 doi: 10.1063/5.0099450
![]() |
[15] |
E. F. D. Goufo, C. Ravichandran, G. A. Birajdar, Self-similarity techniques for chaotic attractors with many scrolls using step series switching, Math. Modell. Anal., 26 (2021), 591–611. https://doi.org/10.3846/mma.2021.13678 doi: 10.3846/mma.2021.13678
![]() |
[16] |
C. Ravichandran, K. Logeswari, A. Khan, T. Abdeljawad, J. F. Gómez-Aguilar, An epidemiological model for computer virus with Atangana-Baleanu fractional derivative, Results Phys., 51 (2023), 106601. https://doi.org/10.1016/j.rinp.2023.106601 doi: 10.1016/j.rinp.2023.106601
![]() |
[17] |
K. S. Nisar, K. Logeswari, V. Vijayaraj, H. M. Baskonus, C. Ravichandran, Fractional order modeling the gemini virus in capsicum annuum with optimal control, Fractal Fract., 6 (2022), 61. https://doi.org/10.3390/fractalfract6020061 doi: 10.3390/fractalfract6020061
![]() |
[18] |
K. Sarkar, S. Khajanchi, J. J. Nieto, Modeling and forecasting the COVID-19 pandemic in India, Chaos, Solitons Fractals, 139 (2020), 110049. https://doi.org/10.1016/j.chaos.2020.110049 doi: 10.1016/j.chaos.2020.110049
![]() |
[19] |
S. Khajanchi, K. Sarkar, Forecasting the daily and cumulative number of cases for the COVID-19 pandemic in India, Chaos: Interdiscip. J. Nonlinear Sci., 30 (2020), 071101. https://doi.org/10.1063/5.0016240 doi: 10.1063/5.0016240
![]() |
[20] |
P. Samui, J. Mondal, S. Khajanchi, A mathematical model for COVID-19 transmission dynamics with a case study of India, Chaos, Solitons Fractals, 140 (2020), 110173. https://doi.org/10.1016/j.chaos.2020.110173 doi: 10.1016/j.chaos.2020.110173
![]() |
[21] |
S. Khajanchi, K. Sarkar, J. Mondal, K. S. Nisar, S. F. Abdelwahab, Mathematical modeling of the COVID-19 pandemic with intervention strategies, Results Phys., 25 (2021), 104285. https://doi.org/10.1016/j.rinp.2021.104285 doi: 10.1016/j.rinp.2021.104285
![]() |
[22] |
R. K. Rai, P. K. Tiwari, S. Khajanchi, Modeling the influence of vaccination coverage on the dynamics of COVID-19 pandemic with the effect of environmental contamination, Math. Methods Appl. Sci., 46 (2023), 12425–12453. https://doi.org/10.1002/mma.9185 doi: 10.1002/mma.9185
![]() |
[23] |
N. Anggriani, M. Z. Ndii, R. Amelia, W. Suryaningrat, M. A. A. Pratama, A mathematical COVID-19 model considering asymptomatic and symptomatic classes with waning immunity, Alexandria Eng. J., 61 (2022), 113–124. https://doi.org/10.1016/j.aej.2021.04.104 doi: 10.1016/j.aej.2021.04.104
![]() |
[24] |
X. Lü, H. W. Hui, F. F. Liu, Y. L. Bai, Stability and optimal control strategies for a novel epidemic model of COVID-19, Nonlinear Dyn., 106 (2021), 1491–1507. https://doi.org/10.1007/s11071-021-06524-x doi: 10.1007/s11071-021-06524-x
![]() |
[25] |
R. K. Rai, S. Khajanchi, P. K. Tiwari, E. Venturino, A. K. Misra, Impact of social media advertisements on the transmission dynamics of COVID-19 pandemic in India, J. Appl. Math. Comput., 68 (2022), 19–44. https://doi.org/10.1007/s12190-021-01507-y doi: 10.1007/s12190-021-01507-y
![]() |
[26] |
G. P. Sahu, J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, J. Math. Anal. Appl., 421 (2015), 1651–1672. https://doi.org/10.1016/j.jmaa.2014.08.019 doi: 10.1016/j.jmaa.2014.08.019
![]() |
[27] |
R. Liu, J. Wu, H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Comput. Math. Methods Med., 8 (2007), 153–164. https://doi.org/10.1080/17486700701425870 doi: 10.1080/17486700701425870
![]() |
[28] | J. Pang, J. A. Cui, An SIRS epidemiological model with nonlinear incidence rate incorporating media coverage, in 2009 Second International Conference on Information and Computing Science, Manchester, UK, 3 (2009), 116–119. https://doi.org/10.1109/ICIC.2009.235 |
[29] |
I. Ghosh, P. K. Tiwari, S. Samanta, I. M. Elmojtaba, N. Al-Salti, J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Math. Biosci., 306 (2018), 160–169. https://doi.org/10.1016/j.mbs.2018.09.014 doi: 10.1016/j.mbs.2018.09.014
![]() |
[30] |
A. K. Misra, A. Sharma, J. B. Shukla, Stability analysis and optimal control of an epidemic model with awareness programs by media, Biosystems, 138 (2015), 53–62. https://doi.org/10.1016/j.biosystems.2015.11.002 doi: 10.1016/j.biosystems.2015.11.002
![]() |
[31] |
C. Maji, F. A. Basir, D. Mukherjee, K. S. Nisar, C. Ravichandran, COVID-19 propagation and the usefulness of awarenessbased control measures: A mathematical model with delay, AIMS Math., 7 (2022), 12091–12105. https://doi.org/10.3934/math.2022672 doi: 10.3934/math.2022672
![]() |
[32] | S. Khajanchi, K. Sarkar, J. Mondal, Dynamics of the COVID-19 pandemic in India, preprint, arXiv: 2005.06286v2. https://doi.org/10.48550/arXiv.2005.06286 |
[33] |
W. Wang, S. Ruan, Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775–793. https://doi.org/10.1016/j.jmaa.2003.11.043 doi: 10.1016/j.jmaa.2003.11.043
![]() |
[34] |
W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58–71. https://doi.org/10.1016/j.mbs.2005.12.022 doi: 10.1016/j.mbs.2005.12.022
![]() |
[35] |
X. Zhang, X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433–443. https://doi.org/10.1016/j.jmaa.2008.07.042 doi: 10.1016/j.jmaa.2008.07.042
![]() |
[36] |
C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361
![]() |
[37] | Shanghai Municipal Health Commission, Prevention and control of COVID-19, 2022. Available from: https://wsjkw.sh.gov.cn/yqfk2020/. |
[38] | National Health Commission of the People's Republic of China, Prevention and control of the COVID-19 epidemic, 2022. Available from: http://www.nhc.gov.cn/xcs/xxgzbd/gzbd_index.shtml. |
[39] | National Bureau of Statistics, China Statistical Yearbook, 2022. Available from: http://www.stats.gov.cn/sj/ndsj/. |
[40] |
J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, Convergence properties of the Nelder–Mead simplex method in low dimensions, SIAM J. Optim., 9 (1998), 112–147. https://doi.org/10.1137/S1052623496303470 doi: 10.1137/S1052623496303470
![]() |
[41] |
S. Bera, S. Khajanchi, T. K. Roy, Stability analysis of fuzzy HTLV-I infection model: A dynamic approach, J. Appl. Math. Comput., 69 (2023), 171–199. https://doi.org/10.1007/s12190-022-01741-y doi: 10.1007/s12190-022-01741-y
![]() |
1. | Changjian Wang, Jiayue Zhu, Global dynamics to a quasilinear chemotaxis system under some critical parameter conditions, 2024, 32, 2688-1594, 2180, 10.3934/era.2024099 |