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.

中文翻译：

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时间分数Keller-Segel系统的非负解

我们建立了具有光滑边界的有界域中Dirichlet边界条件的时间分数Keller-Segel系统的非负弱解的存在。由于所考虑的系统具有交叉扩散项，并且相应的扩散矩阵不是正定的，因此我们首先对系统进行正则化。然后，在适当的初始条件假设下，我们使用Galerkin逼近方法建立系统解的存在性。通过分数阶偏微分方程的紧致性准则证明了解的收敛性。标准论证证明了解的非负性。此外，讨论了具有Neumann边界条件的系统的弱解的存在性。