## Steady state solutions for balanced reaction diffusion systems on heterogeneous domains.(English)Zbl 1064.35514

Semilinear diffractive systems of the type $\begin{gathered} -d_{A_i}\Delta u_i = f_{A_i}(u) \text{ in } A,\\ -d_{B_i}\Delta \tilde u_i = f_{B_i}(\tilde u) \text{ in } B,\\ \tilde u_i = u_i \text{ on } \partial A,\\ d_{A_i}\frac {\partial u_i}{\partial \eta _A} = d_{B_i}\frac {\partial \tilde u_i}{\partial \eta _A} \text{ on } \partial A,\\ \tilde u_i = g_i \text{ on } \partial B \setminus \partial A, \end{gathered}$ $$i=1,...,m$$, are studied. Here $$A$$ and $$B$$ are smooth bounded domains in $$\mathbb R^n$$ such that $$\Omega = \bar A \cup B$$ is a smooth bounded domain, $$A$$ is a strict subdomain of $$\Omega$$, $$d_{A_i},\;d_{B_i} > 0$$, $$g_i$$ are nonnegative and smooth, $$f_A = (f_{A_i}),\;f_B = (f_{B_i})$$ are locally Lipschitz vector fields which are quasi-positive, nearly balanced and polynomial bounded. The existence of a nonnegative solution for the case $$n=2$$ is proved. For the case $$n=3$$, the existence of a nonnegative solution is proved under an additional assumption that $$f_A,\;f_B$$ satisfy a quadratic intermediate sum property. In particular, for $$n=2,3$$, the existence of nonnegative solutions is guaranteed if the system arises from standard balanced quadratic mass action kinetics. The results are applied to two multicomponent chemical models.

### MSC:

 35K57 Reaction-diffusion equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35R05 PDEs with low regular coefficients and/or low regular data 35B45 A priori estimates in context of PDEs