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.
The main purpose of this paper is to study the existence of solutions for the nonlinear fractional partial integrodierential equations with Dirichlet boundary condition. Under suitable assumption the results are established by using the Leray-Schauder xed point theorem and Arzela-Ascoli theorem. An example is provided to illustrate the main result.
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