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Global attractor for a chemotaxis model with prevention of overcrowding. (English) Zbl 1065.35072
The author studies the following system of two parabolic equations modelling the chemotaxis with prevention of overcrowding: \[ \begin{cases} \frac{\partial u}{\partial t}=\nabla.(d_u(q(u)-q'(u)u)\nabla u-uq(u)\chi(v)\nabla v)+uf(u),\\ \frac{\partial v}{\partial t}=d_v\Delta v-g_1(u)-vg_2(v),\end{cases} \tag{1} \] which is considered in a bounded domain of \(\mathbb R^n\) equipped by Robin boundary conditions. Here \(d_u\) and \(d_v\) are positive diffusion coefficients, and \(q(u)\), \(g_1(u)\), \(f(u)\), \(g_2(v)\) and \(\chi(v)\) are assumed to satisfy the assumptions which gives the nondegenerate parabolicity of the problem considered and guarantee that the domain \(V_M:=\{v\geq0,\;\;0\leq u\leq U_M\}\) (for some fixed positive \(U_M\)) is invariant with respect to the flow generated by (1). To be more precise, the last assumption requires: \(g_1(u)\geq0,\;\;f(U_M)\leq 0,\;\;q(U_M)=0\) and, for the nondegeneracy, \(d(\xi):=d_u(q(\xi)-q'(\xi)\xi)>0,\;\;\xi\in[0,U_M].\) Then, under some more technical assumptions on the nonlinearities, the author verifies the existence of a global attractor \(\mathcal A\) for equation (1) in the domain \(V_M\) (equipped with the appropriate topology).

35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
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[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Amann, H., Dynamic theory of quasilinear parabolic systems III. global existence, Math. Z., 202, 219-250, (1989) · Zbl 0702.35125
[3] H. Amann, Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems, in: H.J. Schmeiser, H. Triebel (Eds.), Function spaces, Differential Operators and Nonlinear Analysis, Teubner Texte Math. 133 (1993) 9-126. · Zbl 0810.35037
[4] Biler, P., Local and global solvability of some parabolic systems modelling chemotaxis, Adv. math. sci. appl. nachr., 195, 76-114, (1998)
[5] Brenner, M.P.; Levitov, L.S.; Budrene, E.O., Physical mechanism for chemotactic pattern formation by bacteria, Biphys. J., 74, 1677-1693, (1998)
[6] F.A.C.C.C. Chalub, J.F. Rodrigues, Kinetic Models for chemotaxis with threshold, preprint, 2004.
[7] Cholewa, J.W.; Dlotko, T., Global attractors in abstract parabolic problems, (2000), Cambridge University Press Cambridge · Zbl 1024.35058
[8] Friedman, A., Partial differential equations, (1969), Holt, Rinehart and Winston New York
[9] Grisvard, P., Caractérisation de quelques espaces d’interpolation, Arch. rational mech. anal., 25, 40-63, (1967) · Zbl 0187.05901
[10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, 1981. · Zbl 0456.35001
[11] Herrero, M.A.; Velázquez, J.J.L., A blow-up mechanism for a chemotaxis model, Ann. scuola norm. sup. Pisa, 24, 633-683, (1997) · Zbl 0904.35037
[12] Hillen, T.; Painter, K., A parabolic model with bounded chemotaxis-prevention of overcrowding, Adv. appl. math., 26, 280-301, (2001) · Zbl 0998.92006
[13] T. Hillen, K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart. 10 (2002) 501-543. · Zbl 1057.92013
[14] D. Horstmann, Lyapunov functions and \(L^p\)-estimates for a class of reaction-diffusion systems, Coll. Math. 2001 113-127. · Zbl 0966.35022
[15] Horstmann, D., From 1970 until presentthe keller – segel model in chemotaxis and its consequences I, Jahresber. Deutsch.math.-verein., 105, 103-165, (2003) · Zbl 1071.35001
[16] Keller, E.; Segel, L., Initiation of slime mold aggregation viewed as an instability, J. theor. biol., 26, 399-415, (1970) · Zbl 1170.92306
[17] M.R. Myerscough, P.K. Maini, K.J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol. 60 1-26. · Zbl 1002.92511
[18] Nagai, T., Global existence of solutions to a parabolic system for chemotaxis in two space dimensions, Nonlinear anal., 30, 5381-5388, (1997) · Zbl 0892.35082
[19] Nagai, T., Blow-up of radially symmetric solutions to a chemotaxis system, Adv. math. sci. appl., 5, 581-601, (1995) · Zbl 0843.92007
[20] Osaki, K.; Tsujikawa, T.; Yagi, A.; Mimura, M., Exponential attractor for a chemotaxis-growth system of equations, Nonlinear anal., 51, 119-144, (2002) · Zbl 1005.35023
[21] Osaki, K.; Yagi, A., Finite dimensional attractors for one dimensional keller – segel equations, Funkcial. ekvac., 44, 441-469, (2001) · Zbl 1145.37337
[22] Osaki, K.; Yagi, A., Global existence for a chemotaxis-growth system in \(\mathbb{R}^2\), Adv. math. sci. appl., 12, 587-606, (2002) · Zbl 1054.35111
[23] Patlak, C.S., Random walk with persistence and external bias, Bull. math. biol. biophys., 15, 311-338, (1953), 118 (1995) 219-252 · Zbl 1296.82044
[24] A.B. Potapov, T. Hillen, Metastability in Chemotaxis Models, preprint. · Zbl 1170.35460
[25] Redlinger, R., Existence of global attractor for a strongly coupled parabolic system arising in population dynamics, J. differential equations, 118, 219-252, (1995) · Zbl 0826.35054
[26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. · Zbl 0662.35001
[27] Wang, X., Qualitative behavior of solutions of chemotactic diffusion systemseffects of motility and chemotaxis and dynamics, SIAM J. math. anal., 31, 535-560, (2000) · Zbl 0990.92001
[28] D. Wrzosek, Long time behaviour of solutions to a chemotaxis model with volume filling effect, preprint. · Zbl 1104.35007
[29] Yagi, A., Norm behaviour of solutions to a parabolic system of chemotaxis, Math. japonica, 45, 241-265, (1997) · Zbl 0910.92007
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