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 D t β 0 c ρ + ( − △ ) α 2 ρ + ∇ ⋅ ( ρ B ( ρ ) ) = 0 , where Caputo derivative D t β 0 c ρ models memory effects in time, fractional Laplacian ( − △ ) α 2 ρ represents Lévy diffusion and B ( ρ ) = − s n , γ ∫ R n x − y | x − y | n − γ + 2 ρ ( y ) d y 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 ( S α β ( t ) , T α 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 p c ( R n ) ( p c = n α + γ − 2 ), we construct the global existence. Furthermore, we prove the L 1 integrability and integral preservation when the initial data are in L 1 ( R n ) ∩ L p ( R n ) or L 1 ( R n ) ∩ L p c ( R n ) . Finally, some important properties of the mild solutions including the nonnegativity preservation, mass conservation and blowup behaviors are established. [ABSTRACT FROM AUTHOR]