Asymptotics of Eigenvalues of the Jacobi Matrix of a System of Semilinear Parabolic Equations

We consider the stationary spatially homogeneous solutions of a system of semilinear parabolic equations in a bounded domain with Neumann boundary conditions. It is well known that the stability of such solutions is related to the signs of the real parts of the eigenvalues of the linearized operator composed of the Jacobi matrix of the dynamical system and the differential operator generated by a diffusion process. We obtain the asymptotics of these eigenvalues. We also study the special case in which the diffusion operator is described by matrices containing Jordan blocks, which corresponds to the case of cross diffusion.