Generalized Doubly Parabolic Keller-Segel System with Fractional Diffusion

The Keller-Segel model is a system of partial differential equations that describes the movement of cells or organisms in response to chemical signals, a phenomenon known as chemotaxis. In this study, we analyze a doubly parabolic Keller-Segel system in the whole space $\mathbb{R}^d$, $d\geq 2$, where both cellular and chemical diffusion are governed by fractional Laplacians with distinct exponents. This system generalizes the classical Keller-Segel model by introducing superdiffusion, a form of anomalous diffusion. This extension accounts for nonlocal diffusive effects observed in experimental settings, particularly in environments with sparse targets. We establish results on the local well-posedness of mild solutions for this generalized system and global well-posedness under smallness assumptions on the initial conditions in $L^p(\mathbb{R}^d)$. Furthermore, we characterize the asymptotic behavior of the solution.