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

Abstract

This paper investigates the Cauchy problem of the time‐space fractional parabolic‐elliptic Keller–Segel equation in . The global existence and mass conservation of weak solutions are established. Furthermore, the decay estimates and hyper‐contractive estimates of the weak solutions in for any 1 < r < ∞ are constructed. The existence result is obtained by the classical energy method combined with the generalized strong compactness criteria to time fractional PDEs (partial differential equations). And the proofs of decay estimates and hyper‐contractive estimates are based on the generalized comparison principle with Caputo derivative and the decay behavior of the solutions to the nonlinear fractional differential equation with ν > 0 and γ > 1.