Rocky Mountain Journal of Mathematics

Global Attractors for Cross Diffusion Systems on Domains of Arbitrary Dimension

Hendrik Kuiper and Le Dung

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Rocky Mountain J. Math., Volume 37, Number 5 (2007), 1645-1668.

First available in Project Euclid: 5 November 2007

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Kuiper, Hendrik; Dung, Le. Global Attractors for Cross Diffusion Systems on Domains of Arbitrary Dimension. Rocky Mountain J. Math. 37 (2007), no. 5, 1645--1668. doi:10.1216/rmjm/1194275939.

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  • R.A. Adams, Sobolev spaces, Academic Press, New York, 1975.
  • H. Amann, Dynamic theory of quasilinear parabolic equations I. Abstract evolution equations, Nonlinear Anal. T.M.A. 12 (1988), 895-919.
  • --------, Dynamic theory of quasilinear parabolic systems-III. Global existence, Math. Z. 202 (1989), 219-250.
  • --------, Dynamic theory of quasilinear parabolic equations-II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), 13-75.
  • --------, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function spaces, differential operators and nonlinear analysis, H. Schmeisser and H. Triebel, eds., Teubner-Texte Math. 133, Stuttgart and Leipzig, 1993.
  • S. Childress and J.K. Percus, Nonlinear aspects of chemotaxis, Math. Biosc. 856 (1981), 217-237.
  • Y.S. Choi and R. Lui, Multi-dimensional electrochemistry model, Arch. Rational Mech. Anal. 130 (1995), 315-342.
  • A. Friedman, Partial differential equations, Holt, Rinehart and Winston, New York, 1969.
  • J. Hale, Asymptotic behavior of dissipative systems, Amer. Math. Soc. Math. Surveys Mono. 25, 1988.
  • D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes Math. 840, Springer-Verlag, Berlin, 1981.
  • M. Herrero and J. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177-194.
  • T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, J. Adv. Appl. Math. 26 (2001), 280-301.
  • T. Hillen and K.J. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart. 10 (2002), 501-543.
  • D. Horstmann, Lyapunov functionals and $L^p$ estimates for a class of reaction diffusion systems, Adv. Appl. Math. 26 (2001), 280-301.
  • --------, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, Nonlinear Differential Equations Appl. 8 (2001), 399-423.
  • --------, From $1970$ until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV 105 (2003), 103-165.
  • D. Horstmann, From $1970$ until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresberichte der DMV, (2003).
  • D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, preprint.
  • D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, preprint.
  • T. Ichikawa and Y. Yamada, Some remarks on global solutions to quasilinear parabolic system with cross diffusion, Funk. Ekv. 43 (2000), 285-301.
  • W. Jäger and S. Luckhaus, On explosion of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819-824.
  • E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399-415.
  • H.J. Kuiper, A priori bounds and global existence for a strongly coupled quasilinear parabolic system modeling chemotaxis, Electronic J. Differential Equations 2001 (2001), 1-18.
  • K.H.W. Küfner, Global existence for a certain strongly coupled quasilinear parabolic system in population dynamics, Analysis 15 (1995), 343-357.
  • O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'tseva, Linear and quasilinear equations of parabolic type, AMS Transl. Monographs 23, 1968.
  • D. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microbial Ecol. 22 (1991), 175-185.
  • D. Le, Cross diffusion systems on $n$ spatial dimensional domains, Indiana Univ. Math. J., to appear.
  • --------, On a time dependent chemotaxis system, J. Appl. Comp. Math., to appear.
  • --------, Remark on Hölder continuity for parabolic equations and the convergence to global attractors, Nonlinear Anal. T.M.A. 41 (2000), 921-941.
  • --------, Global attractors and steady state solutions for a class of reaction diffusion systems, J. Differential Equations 147 (1998), 1-29.
  • --------, Coexistence with chemotaxis, SIAM J. Math. Anal. 32 (2000), 504-521.
  • D. Le and H.L. Smith, Steady states of models of microbial growth and competition with chemotaxis, J. Math. Anal. Appl. 229 (1999), 295-318.
  • Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Continuous Dynamical Systems 4 (1998), 193-203.
  • T. Nagai, T. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J. 30 (2000), 463-497.
  • K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funk. Ekv. 44 (2001), 441-469.
  • R. Redlinger, Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics, J. Differential Equations 118 (1995), 219-252.
  • N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), 83-99.
  • P.E. Sobolevskii, On the equations of parabolic type in a Banach space, Trudy Moscov. Mat. Obsc. 10 (1961), 297-350.
  • R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Appl. Math. Sci. 68, Springer-Verlag, New York, 1997.
  • X. Wang, Qualitative behavior of solutions of a chemotactic diffusion system: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal. 31 (2000), 535-560.
  • A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Anal. T.M.A. 21 (1993), 531-556.
  • --------, Norm behavior of solutions to a parabolic system of chemotaxis, Math. J. 45 (1997), 241-265.
  • --------, A priori estimates for some quasilinear parabolic system in population dynamics, Kobe J. Math. 14 (1997), 91-108.