Shrinking vs. expanding: the evolution of spatial support in degenerate Keller–Segel systems

We consider radially symmetric solutions of the degenerate Keller–Segel system {align*} {cases} ∂_t u=∇· (uᵐ⁻¹∇ u - u∇ v),\\ 0=Δ v -μ +u,μ =1/|Ω|∫_Ω u, {cases} {align*} in balls Ω⊂ Rⁿ , n≥ 1 , where m > 1 is arbitrary. Our main result states that the initial evolution of the positivity set of u is essentially determined by the shape of the (nonnegative, radially symmetric, Hölder continuous) initial data u 0 near the boundary of its support Br₁(0)Ω : It shrinks for sufficiently flat and expands for sufficiently steep u 0 . More precisely, there exists an explicit constant Acrit ∈ (0, ∞) (depending only on m, n, R, r₁ and ∫_Ω u₀ ) such that if u₀(x)≤ A(r₁-|x|)¹/m-1 for all |x|∈(r₀, r₁) and some r₀∈(0,r₁) and A < Acrit then there are T > 0 and ζ > 0 such that \\, |x| x ∈ supp u(·, t)\,\≤ r₁ -ζ t for all t∈(0, T) , while if u₀(x)≥ A(r₁-|x|)¹/m-1 for all |x|∈(r₀, r₁) and some r₀ ∈ (0, r₁) and A > Acrit then we can find T > 0 and ζ > 0 such that \\, |x| x ∈ supp u(·, t)\,\≥ r₁ +ζ t for all t∈(0, T) .