一类带logistic源项的趋化方程组解的整体存在性和有界性（英文）
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摘要
本文研究了一类具有logistic源项的趋化方程组解的性质.利用先验估计并结合Neumann热半群的衰减性质,本文证明:当logistic源项中的二次项系数足够大时,方程组的齐次Neumann初边值问题的经典解在边界光滑的三维有界区域上整体存在且一致有界.
The properties of solutions of a class of chemotaxis systems with logistic source are considered.By using prior estimates and decay properties of Neumann heat semi-group,it is proved that there exists a unique global classical solution for the homogenous Neumann initial value problem in three-dimensional bounded domain with smooth boundary if the quadratic coefficients of the logistic source is sufficiently large.
The properties of solutions of a class of chemotaxis systems with logistic source are considered.By using prior estimates and decay properties of Neumann heat semi-group,it is proved that there exists a unique global classical solution for the homogenous Neumann initial value problem in three-dimensional bounded domain with smooth boundary if the quadratic coefficients of the logistic source is sufficiently large.
引文
[1]Li H F,Pang F Q,Wang Y L.Properties of a parabolic system with memory boundary condition[J].J Sichuan Univ:Nat Sci Ed(四川大学学报:自然科学版),2016,53:483.
[2]Chen J Q,Wang Y L,Song X J.Blow-up for a parabolic equation with memory boundary condition[J].J Sichuan Univ:Nat Sci Ed(四川大学学报:自然科学版),2014,51:229.
[3]Deng K,Dong Z.Blow up of a parabolic system with nonlinear memory[J].Dynam Cont Discrete Impulsive Sys,2012,19:729.
[4]Deng K.The role of critical exponents in blow-up theorems:the seuel[J].J Math Anal Appl,2000,243:85.
[5]Keller E F,Segel L A.Initiation of slime mold aggregation viewed as an instability[J].J Theor Biol,1970,26:399.
[6]Bellomo N,Bellouquid A,Tao Y,et al.Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues[J].Math Methods Appl,2015,25:1663.
[7]Winkler M.Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system[J].J Math Pure Appl,2013,100:748.
[8]Osaki K,Tsujikawa T,Yagi A,et al.Exponenitial attactor for a chemotaxis-growth system of equations[J].Nonlinear Anal:Theor,2002,51:119.
[9]Nagai T,Senba T,Yoshida K.Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[J].Funkc Ekvacioj:Ser I,1997,3:411.
[10] Winkler M.Aggregation vs global diffusive behavior in the higher-dimensional Keller-Segel model[J].J Differ Equations,2010,248:2889.
[11] Hillen T,Painter K J.A user's guide to PDF models for chemotaxis[J].J Math Biol,2009,58:183.
[12] Tao Y,Winkler M.Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity[J].J Differ Equations,2012,252:692.
[13] Ishida S,Seki K,Yokota T.Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains[J].J Differ Equations,2014,256:2993.
[14] Yang C,Cao X,Jiang Z,Zheng S.Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source[J].J Math Anal Appl,2015,430:585.
[15] Nakaguchi E,Osaki K.Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation[J].Discrete Cont Dyn B,2013,18:2627.
[16] Herrero M A,Velzquez J J L.A blow-up mechanism for a chemotaxis model[J].Annali Della Scuola Normale Superiore Di Pisa Classe Di Scienze,1997,24:633.
[17] Nagai T.Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains[J].J Inequal Appl,2001,6:37.
[18] Osaki K,Yagi A.Finite dimensional attractors for onedimensional Keller-Segel equations[J].Funkc Ekvacioj-Ser I,2001,44:441.
[19] Winkler M.Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source[J].Commun Part Diff Eq,2010,35:1516.
[20] Winkler M.Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction[J].J Math Anal Appl,2011,384:261.
[21] Haroske D D,Triebel H.Distributions,Sobolev spaces,elliptic equations[M].Zurich:European Mathematical Society,2008.
[22] Mizoguchi N,Souplet P.Nondegeneracy of blow-up points for the parabolic Keller-Segel system[J].Ann Inst H PoincaréAnal Non Linéaire,2014,31:851.
[2]Chen J Q,Wang Y L,Song X J.Blow-up for a parabolic equation with memory boundary condition[J].J Sichuan Univ:Nat Sci Ed(四川大学学报:自然科学版),2014,51:229.
[3]Deng K,Dong Z.Blow up of a parabolic system with nonlinear memory[J].Dynam Cont Discrete Impulsive Sys,2012,19:729.
[4]Deng K.The role of critical exponents in blow-up theorems:the seuel[J].J Math Anal Appl,2000,243:85.
[5]Keller E F,Segel L A.Initiation of slime mold aggregation viewed as an instability[J].J Theor Biol,1970,26:399.
[6]Bellomo N,Bellouquid A,Tao Y,et al.Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues[J].Math Methods Appl,2015,25:1663.
[7]Winkler M.Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system[J].J Math Pure Appl,2013,100:748.
[8]Osaki K,Tsujikawa T,Yagi A,et al.Exponenitial attactor for a chemotaxis-growth system of equations[J].Nonlinear Anal:Theor,2002,51:119.
[9]Nagai T,Senba T,Yoshida K.Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[J].Funkc Ekvacioj:Ser I,1997,3:411.
[10] Winkler M.Aggregation vs global diffusive behavior in the higher-dimensional Keller-Segel model[J].J Differ Equations,2010,248:2889.
[11] Hillen T,Painter K J.A user's guide to PDF models for chemotaxis[J].J Math Biol,2009,58:183.
[12] Tao Y,Winkler M.Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity[J].J Differ Equations,2012,252:692.
[13] Ishida S,Seki K,Yokota T.Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains[J].J Differ Equations,2014,256:2993.
[14] Yang C,Cao X,Jiang Z,Zheng S.Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source[J].J Math Anal Appl,2015,430:585.
[15] Nakaguchi E,Osaki K.Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation[J].Discrete Cont Dyn B,2013,18:2627.
[16] Herrero M A,Velzquez J J L.A blow-up mechanism for a chemotaxis model[J].Annali Della Scuola Normale Superiore Di Pisa Classe Di Scienze,1997,24:633.
[17] Nagai T.Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains[J].J Inequal Appl,2001,6:37.
[18] Osaki K,Yagi A.Finite dimensional attractors for onedimensional Keller-Segel equations[J].Funkc Ekvacioj-Ser I,2001,44:441.
[19] Winkler M.Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source[J].Commun Part Diff Eq,2010,35:1516.
[20] Winkler M.Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction[J].J Math Anal Appl,2011,384:261.
[21] Haroske D D,Triebel H.Distributions,Sobolev spaces,elliptic equations[M].Zurich:European Mathematical Society,2008.
[22] Mizoguchi N,Souplet P.Nondegeneracy of blow-up points for the parabolic Keller-Segel system[J].Ann Inst H PoincaréAnal Non Linéaire,2014,31:851.