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A no-flux initial-boundary value problem for the doubly degenrate parabolic system    $$\begin{aligned} \left\{ \begin{array}{l} u_t = \nabla \cdot \big ( uv\nabla u\big ) + \ell uv, \\ v_t = \Delta v - uv, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$  u t = ∇ · ( u v ∇ u ) + ℓ u v , v t = Δ v - u v , ( ⋆ ) is considered in a smoothly bounded convex domain    $$\Omega \subset \mathbb {R}^n$$  Ω ⊂ R n , with    $$n\ge 1$$  n ≥ 1 and    $$\ell \ge 0$$  ℓ ≥ 0 . The first of the main results asserts that for nonnegative initial data    $$(u_0,v_0)\in (L^\infty (\Omega ))^2$$  ( u 0 , v 0 ) ∈ ( L ∞ ( Ω ) ) 2 with    $$u_0\not \equiv 0$$  u 0 ≢ 0 ,    $$v_0\not \equiv 0$$  v 0 ≢ 0 and    $$\sqrt{v_0}\in W^{1,2}(\Omega )$$  v 0 ∈ W 1 , 2 ( Ω ) , there exists a global weak solution ( u ,  v ) which, inter alia, belongs to    $$C^0(\overline{\Omega }\times (0,\infty )) \times C^{2,1}(\overline{\Omega }\times (0,\infty ))$$  C 0 ( Ω ¯ × ( 0 , ∞ ) ) × C 2 , 1 ( Ω ¯ × ( 0 , ∞ ) ) and satisfies    $$\sup _{t&gt;0} \Vert u(\cdot ,t)\Vert _{L^p(\Omega )}&lt;\infty $$  sup t &gt; 0 ‖ u ( · , t ) ‖ L p ( Ω ) &lt; ∞ for all    $$p\in [1,p_0)$$  p ∈ [ 1 , p 0 ) with    $$p_0:=\frac{n}{(n-2)_+}$$  p 0 : = n ( n - 2 ) + . It is next seen that for each of these solutions one can find    $$u_\infty \in \bigcap _{p\in [1,p_0)} L^p(\Omega )$$  u ∞ ∈ ⋂ p ∈ [ 1 , p 0 ) L p ( Ω ) such that, within an appropriate topological setting,    $$(u(\cdot ,t),v(\cdot ,t))$$  ( u ( · , t ) , v ( · , t ) ) approaches the equilibrium    $$(u_\infty ,0)$$  ( u ∞ , 0 ) in the large time limit. Finally, in the case    $$n\le 5$$  n ≤ 5 a result ensuring a certain stability property of any member in the uncountably large family of steady states    $$(u_0,0)$$  ( u 0 , 0 ) , with arbitrary and suitably regular    $$u_0:\Omega \rightarrow [0,\infty )$$  u 0 : Ω → [ 0 , ∞ ) , is derived. This provides some rigorous evidence for the appropriateness of (   $$\star $$  ⋆ ) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in (   $$\star $$  ⋆ ).

Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar