Hi,

Are there any existence results for the coupled system of linear parabolic PDEs: $$u_t - a_1u_{xx} - a_2u_x - a_3u = f_1$$ $$v_t - a_3u_{xx} - a_4u_x - a_5u - a_6v_{xx} - a_7v_x - a_8v = f_2$$ where the $a_i(x,t)$ are functions with nice properties (eg. continuous). I'm looking for solutions in parabolic Holder space preferably.

I tried Ladyzenskaja but they don't seem to be able to handle this kind of system.

Thank you.