In this paper,we consider the fully parabolic Chemotaxis system with the general logistic source{ut=Δ(γ(v)u)+λu-μu^(k),x∈Ω,t>0,vt=△v+wz,x∈Ω,t>0,wt=-wz,x∈Ω,t>0,zt=△z-z+u,x∈Ω,t>0 whereΩ⊂ℝn(n≥...In this paper,we consider the fully parabolic Chemotaxis system with the general logistic source{ut=Δ(γ(v)u)+λu-μu^(k),x∈Ω,t>0,vt=△v+wz,x∈Ω,t>0,wt=-wz,x∈Ω,t>0,zt=△z-z+u,x∈Ω,t>0 whereΩ⊂ℝn(n≥1)is a smooth and bounded domain,λ≥0,μ≥0,κ>1,and the motility function satisfies thatγ(v)∈C3([0,∞)),γ(v)>0,γ′(v)≤0 for all v≥0.Considering the Neumann boundary condition,we obtain the global boundedness of solutions if one of the following conditions holds:(i)λ=μ=0,1≤nλ3;(ii)λ>0,μ>0,combined withκ>1,1≤n≤3 or k>n+2/4,,n>3.Moreover,we prove that the solution (u, v, w, z) exponentially converges to the constant steady state ((λ/μ)1/k-1,∫Ωv0dx+∫Ωw0dx/|Ω|,0,(λ/μ)1/k-1).展开更多
This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),...This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0 vt=Δv+w−v,x∈Ω,t>0,wt=u−w,x∈Ω,t>0,in a bounded domainΩ⊂R^n(n≥2)with smooth boundary ∂Ω,where the diffusion coefficient D(u)and the chemotactic sensitivity function S(u)are supposed to satisfy D(u)≥M1(u+1)^−αand S(u)≤M2(u+1)^β,respectively,where M1,M2>0 and α,β∈R.Moreover,the logistic source f(u)is supposed to satisfy f(u)≤a−μu^γ with μ>0,γ≥1,and a≥0.Asα+2β<γ−1+2γ/n,we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.展开更多
This work explores the predator-prey chemotaxis system with two chemicals {u_(t) = Δu + ∇ ・ (u∇v) + μ_(1)u(1 − u − α_(1)w), x ∈ Ω, t > 0,v_(t )= Δv −α_(1)v + β_(1)w, x ∈ Ω, t > 0,w_(t) = Δw − ξ∇ ・ (w...This work explores the predator-prey chemotaxis system with two chemicals {u_(t) = Δu + ∇ ・ (u∇v) + μ_(1)u(1 − u − α_(1)w), x ∈ Ω, t > 0,v_(t )= Δv −α_(1)v + β_(1)w, x ∈ Ω, t > 0,w_(t) = Δw − ξ∇ ・ (w∇z) + μ_(2)w(1 + α_(2)u − w), x ∈ Ω, t > 0,z_(t) =Δz −α_(2)z +β_(2)u, x ∈ Ω, t > 0, in an arbitrary smooth bounded domainΩ■R^(n) under homogeneous Neumann boundary conditions.The parameters in the system are positive.We first prove that if n≤3,the corresponding initial-boundary value problem admits a unique global bounded classical solution,under the assumption thatχ,ξ,μ_(i),a_(i),α_(i) andβ_(i)(i=1,2)satisfy some suitable conditions.Subsequently,we also analyse the asymptotic behavior of solutions to the above system and show that·when a_(1)<1 and bothμ1/χ^(2) andμ2/ξ^(2) are sufficiently large,the global solution(u,v,w,z)of this system exponentially converges to(1-a_(1)/1+a_(1)a_(2),β_(1)(1+a_(2))/α_(1)(1+a_(1)a_(2)),1+a_(2)/1+a_(1)a_(2),β_(2)(1-a_(1))/α_(2)(1+a_(1)a_(2)))as t→∞;·when a1>1 andμ_(2)/ξ_(2) is sufficiently large,the global bounded classical solution(u,v,w,z)of this system exponentially converges to(0,α_(1)/β_(1),1,0)as t→∞;·when a1=1 andμ_(2)/ξ_(2) is sufficiently large,the global bounded classical solution(u,v,w,z)of this system polynomially converges to(0,α_(1)/β_(1),1,0)as t→∞.展开更多
In this article,we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation.Based on defining a Finsler-type norm on the tangent space for solutions,we first establish the Li...In this article,we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation.Based on defining a Finsler-type norm on the tangent space for solutions,we first establish the Lipschitz metric for smooth solutions,then by proving the generic regularity result,we extend this metric to general weak solutions.展开更多
基金supported by the NSFC(12301260)the Hong Kong Scholars Program(XJ2023002,2023-078)+14 种基金the Double First-Class Construction-Talent Introduction of Southwest University(SWU-KR22037)the Chongqing Post-Doctoral Fund for Staying in Chongqing(2022)partially supported by the NSFC(12271064,11971082)the Chongqing Talent Support Program(cstc2022ycjh-bgzxm0169)the Natural Science Foundation of Chongqing(cstc2021jcyj-msxmX1051)the Fundamental Research Funds for the Central Universities(2020CDJQY-Z001,2019CDJCYJ001)the Key Laboratory of Nonlinear Analysis and its Applications(Chongqing University)Ministry of EducationChongqing Key Laboratory of Analytic Mathematics and Applicationssupported by the NSFC(12301261)the Scientific Research Starting Project of SWPU(2021QHZ016)the Sichuan Science and Technology Program(2023NSFSC1365)the Nanchong Municipal Government-Universities Scientific Cooperation Project(SXHZ045)supported by the China Scholarship Council(202206050060)the Graduate Research and Innovation Foundation of Chongqing(CYB22044)。
文摘In this paper,we consider the fully parabolic Chemotaxis system with the general logistic source{ut=Δ(γ(v)u)+λu-μu^(k),x∈Ω,t>0,vt=△v+wz,x∈Ω,t>0,wt=-wz,x∈Ω,t>0,zt=△z-z+u,x∈Ω,t>0 whereΩ⊂ℝn(n≥1)is a smooth and bounded domain,λ≥0,μ≥0,κ>1,and the motility function satisfies thatγ(v)∈C3([0,∞)),γ(v)>0,γ′(v)≤0 for all v≥0.Considering the Neumann boundary condition,we obtain the global boundedness of solutions if one of the following conditions holds:(i)λ=μ=0,1≤nλ3;(ii)λ>0,μ>0,combined withκ>1,1≤n≤3 or k>n+2/4,,n>3.Moreover,we prove that the solution (u, v, w, z) exponentially converges to the constant steady state ((λ/μ)1/k-1,∫Ωv0dx+∫Ωw0dx/|Ω|,0,(λ/μ)1/k-1).
基金This work is supported by the Youth Doctor Science and Technology Talent Training Project of Xinjiang Uygur Autonomous Region(2017Q087).
文摘This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+f(u),x∈Ω,t>0 vt=Δv+w−v,x∈Ω,t>0,wt=u−w,x∈Ω,t>0,in a bounded domainΩ⊂R^n(n≥2)with smooth boundary ∂Ω,where the diffusion coefficient D(u)and the chemotactic sensitivity function S(u)are supposed to satisfy D(u)≥M1(u+1)^−αand S(u)≤M2(u+1)^β,respectively,where M1,M2>0 and α,β∈R.Moreover,the logistic source f(u)is supposed to satisfy f(u)≤a−μu^γ with μ>0,γ≥1,and a≥0.Asα+2β<γ−1+2γ/n,we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.
基金supported by the Young Scholars Development Fund of SWPU(202199010087)the Scientific Research Starting Project of SWPU(2021QHZ016)+2 种基金supported by the National Natural Science Foundation of China(11771062and 11971082)the Fundamental Research Funds for the Central Universities(2019CDJCYJ001)Chongqing Key Laboratory of Analytic Mathematics and Applications。
文摘This work explores the predator-prey chemotaxis system with two chemicals {u_(t) = Δu + ∇ ・ (u∇v) + μ_(1)u(1 − u − α_(1)w), x ∈ Ω, t > 0,v_(t )= Δv −α_(1)v + β_(1)w, x ∈ Ω, t > 0,w_(t) = Δw − ξ∇ ・ (w∇z) + μ_(2)w(1 + α_(2)u − w), x ∈ Ω, t > 0,z_(t) =Δz −α_(2)z +β_(2)u, x ∈ Ω, t > 0, in an arbitrary smooth bounded domainΩ■R^(n) under homogeneous Neumann boundary conditions.The parameters in the system are positive.We first prove that if n≤3,the corresponding initial-boundary value problem admits a unique global bounded classical solution,under the assumption thatχ,ξ,μ_(i),a_(i),α_(i) andβ_(i)(i=1,2)satisfy some suitable conditions.Subsequently,we also analyse the asymptotic behavior of solutions to the above system and show that·when a_(1)<1 and bothμ1/χ^(2) andμ2/ξ^(2) are sufficiently large,the global solution(u,v,w,z)of this system exponentially converges to(1-a_(1)/1+a_(1)a_(2),β_(1)(1+a_(2))/α_(1)(1+a_(1)a_(2)),1+a_(2)/1+a_(1)a_(2),β_(2)(1-a_(1))/α_(2)(1+a_(1)a_(2)))as t→∞;·when a1>1 andμ_(2)/ξ_(2) is sufficiently large,the global bounded classical solution(u,v,w,z)of this system exponentially converges to(0,α_(1)/β_(1),1,0)as t→∞;·when a1=1 andμ_(2)/ξ_(2) is sufficiently large,the global bounded classical solution(u,v,w,z)of this system polynomially converges to(0,α_(1)/β_(1),1,0)as t→∞.
基金supported by Chongqing Post-doctoral Innovative Talent Support Progran,the Fundamental Research Funds for the Central Universities(XDJK2020C054)China Postdoctoral Science Foundation(2020M673102)+3 种基金the Natural Science Foundation of Chongqing,China,(cstc2020jcyj-bsh X0071)supported by the Fundamental Research Funds for the Central Universities(2019CDJCYJ001,2020CQJQ-Z001)the NSFC(11771062 and 11971082)Chongqing Key Laboratory of Analytic Mathematics and Applications。
文摘In this article,we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation.Based on defining a Finsler-type norm on the tangent space for solutions,we first establish the Lipschitz metric for smooth solutions,then by proving the generic regularity result,we extend this metric to general weak solutions.
