Existence, uniqueness and L∞-bound for weak solutions of a time fractional Keller-Segel system

Abstract

We study the global existence, uniqueness and L∞-bound for the weak solutions to a time fractional Keller-Segel systems with logistic source∂αu∂tα=Δu−∇⋅u∇v+ua−bu,x∈ℝn,t>00=Δv+u,x∈ℝn,t>0where α ∈ (0,1), a ≥ 0, b > 0 with u(x,0) = u0, v(x,t) is represented by the Newton potentialvxt=1nn−2ωn∫ℝn1x−yn−2uydyWe divide the damping coefficient into different cases and use different methods to prove the existence of weak solutions: (i) when b>1−2n, for any initial value u0 and birth rate a ≥ 0, weak solutions exist globally. (ii) when 0<b≤1−2n, weak solutions have global existence under the condition of small initial data u0 and small birth rate a. Furthermore, by establishing fractional differential inequalities, the L∞-bound of weak solutions is obtained. Finally, we also prove that the weak solution must be unique when the damping effect is strong.