Global solutions for time-space fractional fully parabolic Keller–Segel system

Aruchamy Akilandeeswari; Somnath Gandal; Jagmohan Tyagi

doi:10.4171/zaa/1776

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Global solutions for time-space fractional fully parabolic Keller–Segel system

Aruchamy Akilandeeswari
Anna University, Chennai, India
Somnath Gandal
Indian Institute of Technology Gandhinagar, Gandhinagar, India
Jagmohan Tyagi
Indian Institute of Technology Gandhinagar, Gandhinagar, India

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Abstract

We show the existence of a global solution to time-space fractional fully parabolic Keller–Segel system:

⎩ ⎨ ⎧_{0}^{c} D_{t}^{β} u + (- Δ)^{α /2} u + \nabla \cdot (u \nabla v)_{0}^{c} D_{t}^{β} v + (- Δ)^{α /2} v - u u (x, 0) = u_{0} (x), v (x, 0) = 0, = 0, = v_{0} (x), x \in R^{n}, t > 0, x \in R^{n}, t > 0, x \in R^{n},

under the smallness condition on the initial data, where $0 < β < 1$ , $1 < α \leq 2$ and $n \geq 2$ , $u$ and $v$ denote the cell density and the concentration of the chemoattractant, respectively, and $_{0}^{c} D_{t}^{β}$ denotes the Caputo fractional derivative of order $β$ with respect to time $t$ . The nonlocal operator $(- Δ)^{α /2}$ , defined with respect to the space variable $x$ , is known as the Laplacian of order $\frac{α}{2}$ . We establish the existence of weak solution to the above system by fixed-point arguments under suitable conditions on $u_{0}$ and $v_{0}$ .

Cite this article

Aruchamy Akilandeeswari, Somnath Gandal, Jagmohan Tyagi, Global solutions for time-space fractional fully parabolic Keller–Segel system. Z. Anal. Anwend. (2024), published online first

DOI 10.4171/ZAA/1776