2025/11/16 by Zhang, Zexin
#35B09 #35B40 #35B44 #35J57 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2511.12401
In this paper, we investigate the uniform regularity and asymptotic behavior of solutions to the following Lotka-Volterra type system of strong competition with Dirichlet boundary conditions:
\
-Δui,β = fi,β(x, ui,β) - βui,βpi ∑_\substackj=1
j ≠ ik aij uj,βpj, ui,β · gt; 0 · amp; in Ω,
ui,β = φi,β · amp; on ∂Ω,.
where N ≥ 1, 1 ≤ i ≤ k with k ≥ 3, β> 0, pi ≥ 1, aij > 0 for i ≠ j, and Ω is a C1,Dini bounded domain in ℝN. First, we prove that the uniform boundedness of the solutions implies their uniform interior and global Lipschitz boundedness as β→ +∞. Such uniform results are optimal; partial versions thereof are known in the literature for symmetric coefficients (i.e., aij = aji for all i ≠ j) and homogeneous competition terms (i.e., pi = pj for all i≠ j). Here, we establish an Alt-Caffarelli-Friedman type monotonicity formula for the system and then employ blow-up analysis to show that these results also hold in the asymmetric or nonhomogeneous case. Next, as consequences of the uniform optimal regularity, we derive sharp quantitative pointwise estimates for the densities near the interface between different components.