Resumen de Boundedness and Large Time Behavior for Flux Limitation in a Two-Species Chemotaxis System

Chun Wu, Xiaojie Huang

• In this paper, the following cross-diffusion system is investigated ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ ut =∇· (u + 1)m−1∇u −∇· u∇z (1+|∇z| 2)α , 0 = z − z + v, vt =∇· (v + 1)m−1∇v −∇· v∇w (1+|∇w| 2)α , 0 = w − w + u, in a bounded domain ⊂ Rn (n ≥ 2) with smooth boundary ∂. Under the condition that α > 2n−mn−2 2(n−1) and m ≥ 1, it is shown that the problem possesses a global bounded classical solution. Moreover, we also investigated the large time behavior of the solution, and obtained the corresponding solution exponentially converges to a constant stationary solution when the initial data u0 is sufficiently small.