Smooth Solutions to a Quasi-Linear System of Diffusion Equations for a Certain Population Model

摘要

The system of diffusion equations proposed by Kawasaki, Shigesada and Teramoto describes a population model of two competing species with self- and cross-population pressures. The densities of the two species are denoted by u and v. In this paper the author studies the initial-boundary value problem associated with (*). The Neumann boundary condition (***) corresponds to the case where the flux is zero at the boundary. Many investigators have considered nonlinear diffusion systems arising from various physical and biological problems. These equations, however, have a special structure: the highest order derivatives are not coupled or, at least, the coefficient matrix for the highest order derivatives is positive definite. This is not the case for the system (*) and hence, some of the techniques which are effective for the system are no longer applicable to our case. Nevertheless, we can still use Sobolevski's method (see Reference 2) to establish the local existence of solutions. Under the special assumption c1 = c2 in (*), we can also prove the global existence of solutions by energy estimates. The unusual structure of (*) seems to make it difficult to settle the question of asymptotic stability of solutions.