带有logistic源的生物趋化模型解的全局有界性
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摘要
研究了一个关于两个物种趋化模型的初边值问题{u_t=△u-▽·(uχ_1(w)△w)+μ_1u(1-u),x∈Ω,t>0,vt=△v-▽·(vχ_2(w)△w)+μ_2v(1-v),x∈Ω,t>0,wt=△w+u-w-vw x∈Ω,t>0{,其中ΩR~n(n≥1)是边界光滑的有界区域,χ_i(w)(i=1,2)为趋化敏感函数且满足χ_i(w)≤χ_i/(1+α_iw)~(δ_i),初值u_0,v_0∈C~0(Ω)和w_0∈W~(1,∞)(Ω)且χ_i,α_i,μ_1和μ_2为正,δ_i>1。则当参数槇χ_i和μ_1+μ_2满足一定条件时,表明此模型的初边值问题有唯一的经典解且一致有界。
This paper deals with the global boundedness of the two-species chemotaxis system{u_t= △u- ▽·( uχ_1( w) △w) + μ_1u( 1- u), x∈Ω,t > 0,vt= △v- ▽·( vχ_2( w) △w) + μ_2v( 1- v), x∈Ω,t > 0,wt= △w + u- w- vw x∈Ω,t > 0,under homogeneous Neumann boundary condition in a smoothly bounded domain ΩR~n( n≥1),with nonnegative intial data u_0,v_0∈C~0( Ω) and w_0∈W~(1,∞)( Ω).χ_i,α_i,μ_1has a chemotactic sensitivity function and satisfies χ_i( w) ≤χ_i/( 1 + α_iw)~(δ_i),where the parameters χ_i,α_i,μ_1 and μ_2 are positive δ_i> 1. Under the condition that χ_1,χ_2and μ_1+ μ_2 satisfy some specified conditions,the corresponding initial-boundary value problem possesses a unique global classical solu_tion and is uniformly bounded.
This paper deals with the global boundedness of the two-species chemotaxis system{u_t= △u- ▽·( uχ_1( w) △w) + μ_1u( 1- u), x∈Ω,t > 0,vt= △v- ▽·( vχ_2( w) △w) + μ_2v( 1- v), x∈Ω,t > 0,wt= △w + u- w- vw x∈Ω,t > 0,under homogeneous Neumann boundary condition in a smoothly bounded domain ΩR~n( n≥1),with nonnegative intial data u_0,v_0∈C~0( Ω) and w_0∈W~(1,∞)( Ω).χ_i,α_i,μ_1has a chemotactic sensitivity function and satisfies χ_i( w) ≤χ_i/( 1 + α_iw)~(δ_i),where the parameters χ_i,α_i,μ_1 and μ_2 are positive δ_i> 1. Under the condition that χ_1,χ_2and μ_1+ μ_2 satisfy some specified conditions,the corresponding initial-boundary value problem possesses a unique global classical solu_tion and is uniformly bounded.
引文
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[2]OSAKI K,YAGI A.Finite dimensional attractors for one-dimensional Keller-Segel equations[J].Funkcial.Ekvac.2001,44(3):441-469.
[3]NAGAI T,SENBA T,YOSHIDA K.Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[J].Funkcial.Ekvac.1997,40(3):411-433.
[4]TAO Y,WINKLER M.Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity[J].J.Differential Equations,2012,252:692-715.
[5]CAO X.Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces[J].Discrete Contin.Dynam.Syst.Ser.A,2015,35:1891-1904.
[6]WINKLER M.Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model[J].J.Differential Equations,2010,248(12):2889-2905.
[7]TAO Y,WINKLER M.Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis systemwith consumption of chemoattractant[J].J.Differential Equations,2010,252(3):2520-2543.
[8]LI Y.Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species[J].J.Math.Anal.Appl.,2015,429(2):1291-1304.
[9]CHOI Y S,WANG Z A.Prevention of blow-up by fast diffusion in chemotaxis[J].J.Math.Anal.Appl.,2010,362(2):553-564.
[10]DELGADO M,GAYTE I,MORALES-RODRIGO C,SUREZ A.An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary[J].Nonlinear Anal.,2010,72(1):330-347.
[11]STINNER C,TELLO J I,WINKLER M.Competitive exclusion in a two-species chemotaxis model[J].J.Math.Biol.,2014,68(7):1607-1626.
[12]TEMAM R.Navier-stokes equations,theory and numerical analysis[M].Amsrerdam:North-Holland publishing Co.,1977.
[13]FRIEDMAN A.Partial differential equations[M].New York:Holt,Rinehart and Winston,1969.
[14]WINKLER M,DJIE K C.Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect[J].Nonlinear Anal.,2010,72(2):1044-1064.
[15]ALIKAKOS N D.Lp(Ω)bounds of solution of reaction-diffusion equations[J].Comn.Partial Differential Equations,1979,4(8):827-868.