Bounded small-signal solutions in a three-dimensional doubly degenerate chemotaxis system

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Abstract
A no-flux initial-boundary value problem for { l u t = ∇ ⋅ ( u v ∇ u ) − χ ∇ ⋅ ( u 2 v ∇ v ) + ℓ u v , v t = Δ v − u v ,                                                                                                                               ( ⋆ ) is considered in a smoothly bounded and not necessarily convex domain Ω ⊂ R n with n ⩽ 3 , with nonnegative parameters χ and ℓ , and under the initial condition ( u , v ) | t = 0 = ( u 0 , v 0 ) with given functions u 0 and v 0 from W 1 , ∞ ( Ω ) which satisfy u 0 ⩾ 0 , u 0 ≢ 0 , and v 0 > 0 in Ω ― . It is seen that under an appropriate smallness assumption on v 0 , a global bounded weak solution exists which approaches a nontrivial equilibrium ( u ∞ , 0 ) in the large time limit. This is achieved by means of a Moser iteration involving an apparently novel class of functional inequalities, arranged along with a regularity analysis of signal gradients in a loop-type manner. The estimates gained in the course of this are finally used to indicate that two fundamental dynamical properties known from one-dimensional situations indeed extend to arbitrary domains of physical relevance: It is shown, namely, that if u 0 is not constant and v 0 is subject to a possibly strengthened smallness assumption, then u ∞ ≢ c o n s t . , and a stability property of each of the states ( u ⋆ , 0 ) , with arbitrary nonnegative and nontrivial u ⋆ ∈ W 1 , ∞ ( Ω ) , is identified.
Funding Information
  • Deutsche Forschungsgemeinschaft (462888149)

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