# Coupled system of linear parabolic PDEs

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.

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Can I not solve for $u$ first, then solve for $v$? – PinkPanther Jul 27 '12 at 12:48
Your question makes this sound like homework. – Deane Yang Jul 27 '12 at 13:24
@Deane it's definitely not homework. – PinkPanther Jul 27 '12 at 14:17
I am interested in this as I want to apply an inverse function theorem argument for existence of a nonlinear system. – PinkPanther Jul 27 '12 at 14:17
I don't think you'll get good results if the $a_i$ are merely continuous, since in that case there may not be a good maximal regularity theory. But if all coefficients and the $f_i$ are in $C^{\alpha,\alpha/2}$ in the sense of Ladyzhenskaya, then the mapping $M: f_1 \mapsto a_3u_{xx} + a_4 u_x + a_5 u$ is linear and bounded and therefore you are just solving a linear equation for $v$. – Hans Engler Jul 28 '12 at 1:03