Boundedness and exponential stabilization for time-space fractional parabolic-elliptic Keller-Segel model in higher dimensions

TL;DR

This paper establishes boundedness and exponential stabilization of solutions for a higher-dimensional time-space fractional Keller-Segel model, using fractional inequalities, Lyapunov methods, and Duhamel integral equations.

Contribution

It introduces new techniques for proving exponential stabilization and boundedness in a fractional Keller-Segel model in higher dimensions.

Findings

Solutions are bounded and exponentially stabilize to steady states.

Fractional Duhamel integral equations are effective in analyzing stabilization.

Lyapunov functionals are used to handle the fractional parabolic-elliptic equations.

Abstract

For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{\beta }u=-(-\Delta )^{\frac{\alpha}{2}}(\rho (v)u),& t>0\\ (-\Delta )^{\frac{\alpha}{2}} v+v=u,& t>0 \end{cases} \end{equation*} $x\in\Omega, \Omega \subset \mathbb{R}^{n}, \beta\in (0,1),\alpha\in (1,2)$, we consider for $n\geq 3$ the problem of finding a time-independent upper bound of the classical solution such that as $\theta>0,C>0$ \begin{equation*} \left \| u(\cdot ,t)-\overline{u_{0}} \right \|_{L^{\infty }(\Omega )}+\left \| v(\cdot ,t)-\overline{u_{0}} \right \|_{W^{1,\infty }(\Omega )}\leq Ce^{(-\theta)^{1/\beta}t}, \end{equation*} where $\overline{u_{0}}=\frac{1}{\left | \Omega \right |}\int _{\Omega }u_{0}dx$. We find such solution in the special cases of time-independent upper bound of the concentration with Alikakos-Moser iteration and fractional differential inequality. In those cases the problem is reduced to a time-space fractional parabolic-elliptic equation which is treated with Lyapunov functional methods. A key element in our construction is a proof of the exponential stabilization toward the constant steady states by using fractional Duhamel type integral equation.