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

Abstract We consider radially symmetric solutions of the degenerate Keller–Segel system align* cases ₜ u= (u^m-1 u - u v), \\ 0= v - +u, =1||_ u, cases align* in balls 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 Bₑ䃑 (₀): It shrinks for sufficiently flat and expands for sufficiently steep u 0. More precisely, there exists an explicit constant A₂ₑ₈ₓ (0, ) (depending only on m, n, R, r₁ and _ u₀) such that if u₀ (x) A (r₁-|x|) ¹m-1 for all |x| (r₀, r₁) and some r₀ (0, r₁) and A A₂ₑ₈ₓ then there are T > 0 and ζ > 0 such that \\, |x| x supp u (, t) \, \ r₁ - t for all t (0, T), while if u₀ (x) A (r₁-|x|) ¹m-1 for all |x| (r₀, r₁) and some r₀ (0, r₁) and A A₂ₑ₈ₓ then we can find T > 0 and ζ > 0 such that \\, |x| x supp u (, t) \, \ r₁ + t for all t (0, T).