Existence, uniqueness and L8-bound for weak solutions of a time fractional Keller-Segel system

We study the global existence, uniqueness and L-infinity-bound for the weak solutions to a time fractional Keller-Segel systems with logistic source {& part;(alpha)u/& part;t(alpha) = delta u-& nabla; .(u & nabla;v) + u(a-bu), x is an element of R-n, t > 0 0 = delta v + u, x is an element of R-n, t > 0 where alpha is an element of (0,1), a >= 0, b > 0 with u(x,0) = u(0), v(x, t) is represented by the Newton potential v(x,t) = 1/n(n-2)omega(n)integral 1/R-n|x-y|(n-2) u(y)dy We divide the damping coefficient into different cases and use different methods to prove the existence of weak solutions: (i) when b > 1 - 2/n, for any initial value u0 and birth rate a >= 0, weak solutions exist globally. (ii) when 0 < b <= 1 - 2/n, weak solutions have global existence under the condition of small initial data u(0) and small birth rate a. Furthermore, by establishing fractional differential inequalities, the L-infinity-bound of weak solutions is obtained. Finally, we also prove that the weak solution must be unique when the damping effect is strong. (C) 2022 Elsevier Ltd. All rights reserved.