Existence, uniqueness and L∞-bound for weak solutions of a time fractional Keller-Segel system. (July 2022) - Guo, Liujie; Gao, Fei; Zhan, Hui
Abstract: We study the global existence, uniqueness and L ∞ -bound for the weak solutions to a time fractional Keller-Segel systems with logistic source ∂ α u ∂ t α = Δ u − ∇ ⋅ u ∇ v + u a − bu, x ∈ ℝ n, t > 0 0 = Δ v + u, x ∈ ℝ n, t > 0 where α  ∈ (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 ω n ∫ ℝ n 1 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 u 0 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 ∞ -bound of weak solutions is obtained. Finally, we also prove that the weak solution must be unique when the damping effect is strong. Highlights: We generalize the framework of the unbounded region problem to the research framework of the time fractional Keller-Segel equation. According to the size of the damping coefficient in the model, the global existence of the weak solution is verified by different proof methods. We also verify that when the damping coefficient is strong, the L ∞ -bound of the weak solutions of the model is given by establishing a fractional differential inequality and using the properties of the time fractional equation. Finally, we give the conclusion that when the damping coefficient is strong, as long as the weak solution exists, it must be unique.