ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35K45, 35K57, 35B40, 92C15, 92D25 Zacharias, Klaus, Weierstrass Institute, Berlin, Germany Gajewski, Herbert, Weierstrass Institute, Berlin, Germany On a reaction–diffusion system modelling chemotaxis. Chemotaxis can be modelled by the reaction–diffusion system

U_{t} = ΔU − χ∇ · (U∇V), V_{t} = αΔV − βV + δU.

Here U denotes the population density, V the density of the chemotactic agent. The system is considered for positive initial values U_{0}, V_{0} under homogeneous Neumann conditions on R_{+} × Ω where Ω ⊂ R^{2} is a piecewise smooth domain; α, β, δ, χ are positive constants. The system has the Lyapunov function

F (U, V) = ∫_{Ω} \{\frac{χ}{2δ} (α|∇V |^{2} + βV^{2}) + U(log U − χV)\} dx

i. e. this functional decays along solutions as time increases. If

\frac{δχ∥U_{0}∥_{L_{1}}}{4αΘ} < 1 ,

where Θ is the smallest interior angle formed by the boundary curves of Ω then the Lyapunov function remains bounded from below for all times. Under this smallness condition for the initial values there is a (possibly nontrivial) asymptotic state (U^{⋆}, V^{⋆}) .