# zbMATH — the first resource for mathematics

Global behaviour of a reaction-diffusion system modelling chemotaxis. (English) Zbl 0918.35064
By a change of unknown functions in a partial differential system of reaction-diffusion type, which models chemotaxis, it is obtained the system $u_t= \Delta u-\nabla\cdot(u\nabla v),\quad v_t= \alpha\Delta v-\beta v+ \gamma(u-1)\quad\text{on }\mathbb{R}_+\times \Omega,$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$, $$N\geq 2$$, with piecewise smooth boundary $$\Gamma= \partial\Omega$$; $$\alpha$$, $$\beta$$, $$\gamma$$ are positive constants. This system is completed by the initial and boundary conditions $u(0,.)= u_0,\quad v(0,.)= v_0\quad\text{on }\Omega;\quad \nu\nabla u= \nu\nabla u= 0,$ on $$\mathbb{R}_+\times\Gamma$$, where $$\nu$$ is the outer normal on $$\Gamma$$, so it models the dynamics of a population (concentration $$u$$) moving in $$\Omega$$, driven by gradient of chemotactic agents (concentration $$v$$) produced by the population.
The existence and uniqueness of a local weak solution is proved. The main aim of this paper is the study of the global behavior of solutions, using two Lyapunov functionals.

##### MSC:
 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 92C40 Biochemistry, molecular biology 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 92C15 Developmental biology, pattern formation 92D25 Population dynamics (general)
##### Keywords:
local weak solution; Lyapunov functionals
Full Text:
##### References:
 [1] Adams, Sobolev Spaces (1975) [2] Brézis, Mathematics Studies 5, in: Opérateurs Maximaux Monotones et Semi - Groupes de Contractions dans les Espaces de Hilbert (1973) · Zbl 0252.47055 [3] Chang, Conformal Deformation of Metrics on S2, J. Differential Geometry 27 pp 259– (1988) [4] Dore, Lecture Notes in Mathematics 1540 pp 25– (1993) [5] Diaz, Symmetrization in a Parabolic - Elliptic System Related to Chemotaxis, Advances in Mathematical Sciences and Applications 5 pp 659– (1995) · Zbl 0859.35004 [6] Gajewski, On the Basic Equations for Carrier Transport in Semiconductors, Journal of Mathematical Analysis and Applications 113 pp 12– (1986) · Zbl 0642.35038 [7] Gajewski, Zur Konvergenz eines Iterationsverfahrens für Evolutionsgleichungen, Math. Nachr. 68 pp 331– (1975) · Zbl 0334.65045 [8] Gajewski, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (1974) · Zbl 0289.47029 [9] Gajewski, Preprint No. 70, in: About Loss of Regularity and ”blow up” of Solutions for Quasilinear Parabolic Systems (1993) [10] Giusti, Monographs in Mathematics 80 (1984) [11] Gröger, W1,p-Estimates of Solutions to Evolution Equations Corresponding to Nonsmooth Second Order Elliptic Differential Operators, Nonlinear Analysis, Theory, Methods&Applications 18 pp 569– (1992) [12] Gröger, A W1,p -Estimate for Solutions to Mixed Boundary Value Problems for Second Order Elliptic Differential Equations, Math. Annalen 283 pp 679– (1989) [13] Gilbarg, Elliptic Partial Differential Equations of Second Order (Second Edition) (1983) · Zbl 0562.35001 [14] Taxis and Behaviour, Elementary Sensory Systems in Biology, Receptors and Recognition, Series B 5 (1978) [15] Henry, Lecture Notes in Mathematics, No. 840, in: Geometric Theory of Semilinear Parabolic Equations (1981) · Zbl 0456.35001 [16] Herrero , M. A. Medina , E. Velazquez , J. J. L. 1996 [17] Herrero, Singularity Patterns in a Chemotaxis Model Math., Annalen 306 pp 583– (1996) · Zbl 0864.35008 [18] Herrero , M. A. Velazquez , J. J. L. A Blow Up Mechanism for a Chemotaxis Model Preprint Departamento de Matemática Aplicada Universidad Complutense Madrid [19] Herrero , M. A. Velazquez , J. J. L. [20] Jäger, On Explosions of Solutions to a System of Partial Differential Equations Modelling Chemotaxis, Transactions AMS 329 pp 819– (1992) · Zbl 0746.35002 [21] Kazdan, Curvature Functions for Compact 2-Manifolds, Ann. Math. 99 pp 14– (1974) · Zbl 0273.53034 [22] Kufner, Function Spaces (1977) [23] Keller, Initiation of Slime Mold Aggregation Viewed as an Instability, J. Theoret. Biol. 26 pp 399– (1970) · Zbl 1170.92306 [24] Kinderlehrer, An Introduction to Variational Inequalities and Their Applications (1980) · Zbl 0457.35001 [25] Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linèaires (1969) [26] Lions, Problèmes aux Limites non Homogènes 1 (1968) · Zbl 0235.65074 [27] Ladyženskaja , O. A. Solonnikov , V. A. Ural’ceva , N. N. Linear and Quasilinear Equations of Parabolic Type Nauka, Moscow, 1967 (Russian). English translation: Tranal. Math. Monographs 23 1991 [28] Ladyženskaja, Linear and Quasilinear Elliptic Equations (1964) [29] Moser, Dynamical Systems pp 273– (1973) [30] Moser, A Sharp Form of an Inequality by N., Trudinger, Indiana Math. J. 20 pp 1077– (1971) [31] Murray, Mathematical Biology (1989) · Zbl 0682.92001 [32] Schaaf, Stationary Solutions of Chemotaxis Systems, Transactions AMS 292 pp 531– (1985) · Zbl 0637.35007 [33] Trudinger, On Imbeddings into Orlicz Spaces and Some Applications, Journal of Mathematics and Mechanics 17 pp 473– (1967) · Zbl 0163.36402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.