Can a chemotaxis-consumption system recover from a measure-type aggregation state in arbitrary dimension?

We consider the chemotaxis-consumption system \[ ( ⋆ ) { u t a m p ; = Δ u − χ ∇ ⋅ ( u ∇ v ) v t a m p ; = Δ v − u v {cases} u_t &= Δ u - χ ∇ · (u∇ v) \\ v_t &= Δ v - uv {cases} {} \] in a smooth bounded domain Ω ⊆ R n Ω ⊆ R^n , n ≥ 2 n ≥ 2 , with parameter χ > 0 χ > 0 and Neumann boundary conditions. It is well known that, for sufficiently smooth nonnegative initial data and under a smallness condition for the initial state of v v , solutions of the above system never blow up and are even globally bounded. Going in a sense a step further in this paper, we ask the question whether the system can even recover from an initial state that already resembles measure-type blowup. To answer this, we show that, given an arbitrarily large positive Radon measure u 0 u_0 with u 0 ( Ω ¯ ) > 0 u_0(Ω ̄) > 0 as the initial data for the first equation and a nonnegative L ∞ ( Ω ) L^∞ (Ω ) function v 0 v_0 with \[ 0 > ‖ v 0 ‖ L ∞ ( Ω ) > π χ 2 n 0 > \|v_0\|_{L∞(Ω )} > π /χ √2/n \] as initial data for the second equation, it is still possible to construct a global classical solution to the above system. Notably, the above condition on the initial data appears to be weaker than those required in all previous works on ( ⋆ ) even in frameworks of smooth initial data.