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

Zexin Zhang

TL;DR

This work advances the theory of strongly competing Lotka-Volterra type elliptic systems by treating asymmetric and nonhomogeneous interactions under Dirichlet boundary conditions. It introduces an Alt-Caffarelli-Friedman type monotonicity formula tailored to nonhomogeneous systems and develops a novel blow-up framework to obtain uniform interior and global Lipschitz regularity without symmetry or homogeneity assumptions. The authors derive sharp pointwise density estimates near interfaces, establish uniform upper bounds, and characterize asymptotics near regular points and interfaces as the competition parameter $\beta_n\to\infty$, including precise decay rates for nonactive components. These results extend the symmetric/homogeneous literature to asymmetric cases and provide rigorous tools for understanding spatial segregation and free-boundary structure in multi-species diffusion systems with strong competition.

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} -Δu_{i,β} = f_{i,β}(x, u_{i,β}) - βu_{i,β}^{p_i} \sum_{\substack{j=1 \\ j \neq i}}^k a_{ij} u_{j,β}^{p_j}, \quad u_{i,β} > 0 & \text{in } Ω, u_{i,β} = \varphi_{i,β} & \text{on } \partialΩ, \end{array}\right. \end{align*} where $N \geq 1$, $1 \leq i \leq k$ with $k \geq 3$, $β> 0$, $p_i \geq 1$, $a_{ij} > 0$ for $i \neq j$, and $Ω$ 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 $β\to +\infty$. 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.

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

TL;DR

This work advances the theory of strongly competing Lotka-Volterra type elliptic systems by treating asymmetric and nonhomogeneous interactions under Dirichlet boundary conditions. It introduces an Alt-Caffarelli-Friedman type monotonicity formula tailored to nonhomogeneous systems and develops a novel blow-up framework to obtain uniform interior and global Lipschitz regularity without symmetry or homogeneity assumptions. The authors derive sharp pointwise density estimates near interfaces, establish uniform upper bounds, and characterize asymptotics near regular points and interfaces as the competition parameter , including precise decay rates for nonactive components. These results extend the symmetric/homogeneous literature to asymmetric cases and provide rigorous tools for understanding spatial segregation and free-boundary structure in multi-species diffusion systems with strong competition.

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} -Δu_{i,β} = f_{i,β}(x, u_{i,β}) - βu_{i,β}^{p_i} \sum_{\substack{j=1 \\ j \neq i}}^k a_{ij} u_{j,β}^{p_j}, \quad u_{i,β} > 0 & \text{in } Ω, u_{i,β} = \varphi_{i,β} & \text{on } \partialΩ, \end{array}\right. \end{align*} where , with , , , for , and is a bounded domain in . First, we prove that the uniform boundedness of the solutions implies their uniform interior and global Lipschitz boundedness as . Such uniform results are optimal; partial versions thereof are known in the literature for symmetric coefficients (i.e., for all ) and homogeneous competition terms (i.e., for all ). 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.

Paper Structure

This paper contains 13 sections, 27 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An Alt-Caffarelli-Friedman type monotonicity formula
  4. Uniform interior Lipschitz bounds
  5. Asymptotics of the blow-up sequence
  6. Applications of the monotonicity formula
  7. Uniform global Lipschitz bounds
  8. Asymptotic behavior
  9. Uniform upper bound
  10. Asymptotic behavior near regular points
  11. Sharp estimates near the interface $\Gamma_n$
  12. Existence of convergent point sequences
  13. Properties of elements of $\mathscr{S}(\Omega)$ in one dimensional case

Key Result

Theorem 1.1

Let $N \geq 1$ and $\Omega$ be a domain in $\mathbb{R}^N$ (not necessarily proper, bounded, or smooth). Let $k \geq 3$, $p_i \geq 1$, and $a_{ij} > 0$ for all $1 \leq i, j \leq k$ with $i \neq j$. For each $n \in \mathbb{N}$, let $\bm{u}_n \in H^1_{\text{loc}}(\Omega; \mathbb{R}^k) \cap C_{\text{loc where $\beta_{ij;n} = \beta_{ji;n} > 0$ for all $i \neq j$, and $\bm{f}_n$ satisfies condition (H1)

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Lemma 2.1
  • proof
  • ...and 47 more