Partial smoothing effects of local mild solutions of the Keller–Segel system with logistic growth in Besov spaces
Year: 2025
Type: article
Abstract: We consider the Keller–Segel system of parabolic-elliptic type with logistic growth 𝑢 − |𝑢|^<𝜅−𝑙> 𝑢^𝑙 in the whole space ℝ^𝑛, where 𝑛 ≥ 3, 1 < 𝜅 < 2, and 𝑙 ∈ {0, 1}. We show the existence and unique- ness of local mild solutions 𝑢 for initial data 𝑎 ∈ 𝐵^<-2+𝑛∕𝑝>_<𝑝,𝑞> (ℝ^𝑛) under the conditions 𝑛∕2 < 𝑝 < 𝑛, 1 ≤ 𝑞 ≤ ∞, and 2𝑝∕𝑛 < 𝜅 < 2. In addition, partial smoothing effects of the mild solutions 𝑢 are investigated. More precisely, we show that 𝑢 satisfy the original system in a classical sense and have the property 𝑢 ∈ 𝐶^<𝜅+1>_ ((0, 𝑇 ]; 𝐿^∞(ℝ^𝑛)) ∩ 𝐿^∞_ ((0, 𝑇 ]; 𝐶^<𝜅+2>(ℝ^𝑛)). According to the regularities of the term |𝑢|^<𝜅−𝑙> 𝑢^𝑙 with the power nonlinearity, such regularities for 𝑢 seem to be optimal under the general framework beyond the physical sense.
Source: Nonlinear Analysis
Author: Taiki Takeuchi
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,690
OpenAlex ID: W4417490263