Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

Abstract

One of the most important systems for understanding chemotactic aggregation is the Keller-Segel system. We consider the time-fractional Keller-Segel system of order ∈ (0, 1). We prove an existence result with small initial data in a class of Besov-Morrey spaces. Self-similar solutions are obtained and we also show an asymptotic behaviour result. K E Y W O R D SBesov-Morrey, chemotaxis aggregation, fractional derivative, Keller-Segel model, spaces M S C ( 2 0 1 0 ) 26A33, 35A01, 35B40, 35K45, 35K55, 35Q92, 35R11, 92C15, 92C17

On a time fractional diffusion with nonlocal in time conditions

“…By replacing many differential operators of fractional order with different type of PDEs of integer order, we formulate various types of boundary value problems with fractional order. Let us refer to many papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Weak solutions to the Cauchy problem of fractional time‐space Keller–Segel equation

“…26 Moreover, the study on the blow-up asymptotic behavior of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in R d (d ≥ 3) can be found in Souplet and Winkler. 27 Recently, there are some results 1,17,[28][29][30][31][32][33][34][35] involved in the study of the fractional Keller-Segel equation. For example, Escudero 28 constructed the global in time solutions for the fractional diffusion 1 < 𝛼≤2.…”

“…Recently, there are some results 1,17,28‐35 involved in the study of the fractional Keller–Segel equation. For example, Escudero 28 constructed the global in time solutions for the fractional diffusion 1 < α ≤2.…”

Nonnegative solutions to time fractional Keller–Segel system

“…We point out that there are very few papers dealing with time fractional Keller-Segel systems. Recently, (1.1) with d 1 = d 2 = = = = = 1 and (u, v) = u has been studied by Azevedo et al 39 They discussed the global existence of solutions of the fractional Keller-Segel system in R n , n ≥ 2 with small initial data in a class of Besov-Morrey spaces, that is,…”

“…We point out that there are very few papers dealing with time fractional Keller–Segel systems. Recently, () with and χ ( u , v ) = u has been studied by Azevedo et al 39 They discussed the global existence of solutions of the fractional Keller–Segel system in with small initial data in a class of Besov–Morrey spaces, that is, , with the help of iteration method and also obtained self‐similar solutions of the system. Cuevas et al 40 considered the same system with initial conditions , N ≥ 2, and and established the well‐posedness of a solution to the problem.…”