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?
1 Answer
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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).
