Let's consider the parabolic system
$$
\begin{cases}
u\_t - \Delta u -a\Delta(uv) = 0 \\
v\_t - \Delta v - b\Delta(uv) = 0
\end{cases}
$$
with $a,b >0$. What is the name of this system? Are there known results about existence and uniqueness?



Parabolic system with coupling in the diffusion
Such systems belong to the family of **cross diffusion systems**.
For $W^{1,p}$ data ($p$ greater than the dimension), local existence and uniqueness stems from Amann's theorem (look for *Dynamic theory of quasilinear parabolic systems paper* from 1989). For weak solutions (in bounded domains), you can look at the work of Chen and Jungel *Analysis of a parabolic cross-diffusion population model without self-diffusion* in *J. Diff. Eq.* (and authors citing it) based on the remark that and adapted LlogL entropy is dissipated along positive solutions. In population dynamics they have been introduced by Shigesada Kawasaki and Terramoto (*Spatial segregation of interacting species*, *J. Theor. Biol* 1979).
I am far from being exhaustive, there exists a huge literature on those systems (and more complicated versions).


