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In this paper we investigate a class of two-species invasion models with a free boundary and with cross-diffusion and self-diffusion in interval [ 0 , + ∞ ) , where the native species occupies the whole environment [ 0 , + ∞ ) , and the invasion species invade into the habitat of the native species. The systems under consideration are strongly coupled, and the position of the free boundary is determined by the Stefan condition. The aim of this paper is to show the existence of a global classical solution for the free boundary problems by using suitable transformations, the contraction mapping theorem and various estimates. Applications are given to the classical competition and predator-prey models with cross-diffusion and self-diffusion.

This paper deals with a one-dimensional parabolic cross-diffusion system describing two-species models. The system under consideration is strongly coupled and the coefficients of the equations are allowed to be discontinuous. The existence and uniqueness of nonnegative global solutions are given when one of the cross-diffusion pressures is zero. Our approach to the problem is by an approximation method and various estimates.

In this paper, we discuss the n-dimensional diffraction problem for weakly coupled quasilinear parabolic system on a bounded domain Ω, where the interfaces ( ) are allowed to intersect with the outer boundary ∂ Ω and the coefficients of the equations are allowed to be discontinuous on the interfaces. The aim is to show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces, which do not intersect with ∂ Ω. MSC:35R05, 35K57, 35K65.