Estimates in the Generalized Morrey Spaces for Linear Parabolic Systems

The purpose of this paper is to study the parabolic system $u_t^i-D_{\alpha }(a_{ij}^{\alpha \beta }{D}_{\beta }{u}^j)=-div{f}^i$ in the generalized Morrey Space $L_{\varphi }^{2,\lambda }$ . We would like to understand the regularity of the solutions of this system. It will be shown that 1: if $a_{ij}^{\alpha \beta }\in C(\overline{Q_T})$ then $Du\in L_{\varphi }^{2,\lambda }$, and 2: if $a_{ij}^{\alpha \beta }\in VMO(Q_T)$ then $Du\in L_{\varphi }^{2,\lambda }$. Moreover we will be able to obtain estimates on the gradient of the solutions to the system, which will tell us about the regularity of the solutions.