Optimal uniform regularity and asymptotic behavior of solutions to Lotka-Volterra type systems with strong competition and asymmetric coefficients

TL;DR

This paper studies the regularity and asymptotic behavior of solutions to a class of strongly competitive Lotka-Volterra systems with asymmetric coefficients, establishing uniform bounds and sharp estimates using advanced analytical techniques.

Contribution

It extends uniform regularity results to asymmetric and nonhomogeneous systems, employing a new monotonicity formula and blow-up analysis.

Findings

Solutions are uniformly bounded and Lipschitz continuous as competition parameter grows

Established an Alt-Caffarelli-Friedman type monotonicity formula for asymmetric systems

Derived sharp pointwise estimates near interfaces between components

Abstract

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: \begin{align*} \left\{ \begin{array}{ll} -\Delta u_{i,\beta} = f_{i,\beta}(x, u_{i,\beta}) - \beta u_{i,\beta}^{p_i} \sum_{\substack{j=1 \\ j \neq i}}^k a_{ij} u_{j,\beta}^{p_j}, \quad u_{i,\beta} > 0 & \text{in } \Omega, u_{i,\beta} = \varphi_{i,\beta} & \text{on } \partial\Omega, \end{array}\right. \end{align*} where $N \geq 1$, $1 \leq i \leq k$ with $k \geq 3$, $\beta > 0$, $p_i \geq 1$, $a_{ij} > 0$ for $i \neq j$, and $\Omega$ is a $C^{1,\text{Dini}}$ bounded domain in $\mathbb{R}^N$. First, we prove that the uniform boundedness of the solutions implies their uniform interior and global Lipschitz boundedness as $\beta \to +\infty$. Such uniform results are optimal; partial versions thereof are known in the literature for symmetric coefficients (i.e., $a_{ij} = a_{ji}$ for all $i \neq j$) and homogeneous competition terms (i.e., $p_i = p_j$ for all $i\neq j$). 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.