Cauchy problems for Keller–Segel type time–space fractional diffusion equation

This paper investigates Cauchy problems for nonlinear fractional time–space generalized Keller–Segel equation Dtβ0cρ+(−△)^{[Formula presented]}ρ+∇⋅(ρB(ρ))=0, where Caputo derivative Dtβ0cρ models memory effects in time, fractional Laplacian (−△)^{[Formula presented]}ρ represents Lévy diffusion and B(ρ)=−sn,γ∫Rⁿ[Formula presented]ρ(y)dy is the Riesz potential with a singular kernel which takes into account the long rang interaction. We first establish L^r−L^q estimates and weighted estimates of the fundamental solutions (P(x,t),Y(x,t)) (or equivalently, the solution operators (Sα^β(t),Tα^β(t))). Then, we prove the existence and uniqueness of the mild solutions when initial data are in L^p spaces, or the weighted spaces. Similar to Keller–Segel equations, if the initial data are small in critical space L^pc(Rⁿ) (pc=[Formula presented]), we construct the global existence. Furthermore, we prove the L¹ integrability and integral preservation when the initial data are in L¹(Rⁿ)∩L^p(Rⁿ) or L¹(Rⁿ)∩L^pc(Rⁿ). Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established.

Duke Scholars

Author Jian-Guo Liu Physics