Steady state solutions for balanced reaction diffusion systems on heterogeneous domains. (Q1854092)

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.