Estimates in generalized Morrey spaces for linear parabolic systems

In this study, we consider the parabolic system uti−Dα(aijαβDβuj)=−divfi in the generalized Morrey Space Lφ2,λ, where we aim to understand the regularity of the solutions to this system. We show that: (1) if aijαβ∈C(QT‾), then Du∈Lφ2,λ; and (2) if aijαβ∈VMO(QT), then Du∈Lφ2,λ. Moreover, we obtain estimates of the gradient for the solutions to the system, thereby showing the regularity of the solutions.