I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u\_t - a\_1u\_{xx} - a\_2u\_x - a\_3u -a\_4v = f$$
$$v\_t - a\_5u\_{xx} - a\_6u\_x - a\_7u - a\_8v\_{xx} - a\_9v\_x - a\_{10}v = g$$
where the coefficients $a\_i(x,t)$ are in a parabolic version of Holder space (say $C^{k, \alpha}([0,1]\times[0,t])$.


Is there already literature where this is treated? I tried Ladyschenkaja but there this type of system is not present (I believe in that book, they require $a\_1 \equiv a\_5$). I would appreciate if anyone had any references to this problem.


I believe though that I can probably apply a fixed-point argument to this -- again I would appreciate if someone gave a pointer as where such things are discussed. Thanks.



Linear coupled parabolic PDE system with Holder continuous coefficients
There is a classical counterexample due to Plis of an elliptic differential operator with Hölder continuous coefficients without Cauchy uniqueness. This was refined with a counterexample in divergence form by Miller in a 1974 Arch. Rat. Mech.(vol. 54) article for the elliptic and parabolic case.


Hölder continuity is not enough to get uniqueness results for parabolic or elliptic equations.


