| Abstract: |
| This talk is concerned with the Dirichlet problem of a Lotka-Volterra competition system with cross-diffusion terms. The purpose is to derive the limiting behavior of positive steady-state solutions as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The main result shows that almost all limits of positive steady-states can be classified into two types: small coexistence or complete segregation, and the limit of small coexistence can be characterized by positive solutions to a single elliptic equation, while the limit of complete segregation can be characterized by nodal solutions to another single elliptic equation. By the perturbation of solutions of these two limiting equations, we construct the set of positive solutions when equal two cross-diffusion coefficients are sufficiently large. From the viewpoint of the bifurcation diagram, the branch of the small coexistence bifurcates from the trivial solution, and many branches of the segregation bifurcate from solutions on the branch of the small coexistence. |
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