Nonnegative solutions to time fractional Keller–Segel system

We establish the existence of nonnegative weak solutions to time fractional Keller–Segel system with Dirichlet boundary condition in a bounded domain with smooth boundary. Since the considered system has a cross-diffusion term and the corresponding diffusion matrix is not positive definite, we first regularize the system. Then under suitable assumptions on the initial conditions, we establish the existence of solutions to the system by using the Galerkin approximation method. The convergence of solutions is proved by means of compactness criteria for fractional partial differential equations. The nonnegativity of solutions is proved by the standard arguments. Furthermore, the existence of the weak solution to the system with Neumann boundary condition is discussed.