Key points are not available for this paper at this time.
This work is concerned with the doubly degenerate cross-diffusion system (*) u t = (u v u x) x − (u 2 v v x) x + u v, v t = v x x − u v, equation {* \ {arrayl uₜ = (uvuₓ) ₓ - (u² vvₓ) ₓ + uv, \\1mm vₜ = vₗₗ-uv, array. equation that has been proposed as a model for experimentally observable quite complex pattern formation phenomena in bacterial populations. It is shown that for any initial data satisfying adequate regularity and positivity assumptions, a no-flux initial-boundary value problem for the above in a bounded real interval possesses a global weak solution which is continuous in its first and essentially smooth in its second component. This solution is seen to asymptotically stabilize in the sense that (**) u (⋅, t) → u ∞ and v (⋅, t) → 0 as t → ∞ equation ** u (, t) u_ and v (, t) 0 as t equation with some nonnegative <mml: math xmlns: mml="http: //www. w3. org/1998/Math/MathML" alttext="u Subscript normal infinity Baseline element-of upper C Superscript 0 Baseline left-parenthesis normal upper
Michael Winkler (Wed,) studied this question.