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Radially symmetric transition-layer solutions in mass-conserving reaction-diffusion systems with bistable nonlinearity

Xiaoqing He, Quan-Xing Liu, Dong Ye

TL;DR

This work proves the existence of radially symmetric stationary transition-layer solutions to mass-conserving reaction-diffusion systems with bistable nonlinearity on an $N$-ball, under a prescribed total mass. It combines a nonlocal reformulation with a refined matched-asymptotic construction to produce arbitrarily high-order outer/inner expansions and uniform error estimates, then establishes a uniform spectral theory for the linearized operator and applies a Lyapunov–Schmidt reduction to obtain exact solutions close to the approximations. The key contributions include a rigorous high-dimensional transition-layer theory for MCRD systems, explicit characterization of the interface location $R_*(M)$, and quantitative control of residuals and stability-related spectral data. These results provide a rigorous mathematical foundation for pattern formation and phase-separation phenomena in biological and ecological contexts, connecting sharp-interface limits to multiscale mass-conserving dynamics.

Abstract

Mass-conserving reaction-diffusion (MCRD) systems are widely used to model phase separation and pattern formation in cell polarity, biomolecular condensates, and ecological systems. Numerical simulations and formal asymptotic analysis suggest that such models can support stationary patterns with sharp internal interfaces. In this work, we establish for a general class of bistable MCRD systems the existence of nonconstant radially symmetric stationary solutions with a single internal transition layer on an $N$-dimensional ball, for general spatial dimension $N$. Our approach incorporates the global mass constraint directly into a refined matched-asymptotic framework complemented by a uniform spectral/linear analysis. Beyond mere existence, our framework yields arbitrarily high-order asymptotic approximations of the constructed solutions together with quantitative uniform error estimates, which provides a quantitative higher-dimensional theory of transition-layer patterns in MCRD systems and a rigorous justification for their use in modeling phase separation and pattern formation in biological and ecological settings.

Radially symmetric transition-layer solutions in mass-conserving reaction-diffusion systems with bistable nonlinearity

TL;DR

This work proves the existence of radially symmetric stationary transition-layer solutions to mass-conserving reaction-diffusion systems with bistable nonlinearity on an -ball, under a prescribed total mass. It combines a nonlocal reformulation with a refined matched-asymptotic construction to produce arbitrarily high-order outer/inner expansions and uniform error estimates, then establishes a uniform spectral theory for the linearized operator and applies a Lyapunov–Schmidt reduction to obtain exact solutions close to the approximations. The key contributions include a rigorous high-dimensional transition-layer theory for MCRD systems, explicit characterization of the interface location , and quantitative control of residuals and stability-related spectral data. These results provide a rigorous mathematical foundation for pattern formation and phase-separation phenomena in biological and ecological contexts, connecting sharp-interface limits to multiscale mass-conserving dynamics.

Abstract

Mass-conserving reaction-diffusion (MCRD) systems are widely used to model phase separation and pattern formation in cell polarity, biomolecular condensates, and ecological systems. Numerical simulations and formal asymptotic analysis suggest that such models can support stationary patterns with sharp internal interfaces. In this work, we establish for a general class of bistable MCRD systems the existence of nonconstant radially symmetric stationary solutions with a single internal transition layer on an -dimensional ball, for general spatial dimension . Our approach incorporates the global mass constraint directly into a refined matched-asymptotic framework complemented by a uniform spectral/linear analysis. Beyond mere existence, our framework yields arbitrarily high-order asymptotic approximations of the constructed solutions together with quantitative uniform error estimates, which provides a quantitative higher-dimensional theory of transition-layer patterns in MCRD systems and a rigorous justification for their use in modeling phase separation and pattern formation in biological and ecological settings.
Paper Structure (16 sections, 24 theorems, 287 equations)

This paper contains 16 sections, 24 theorems, 287 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Mass-conserving reaction-diffusion system
  3. Assumptions and statement of main result
  4. Construction of the approximate solutions
  5. Preliminaries
  6. Statements on existence of approximate solutions
  7. Outer and inner expansions
  8. Properties of outer expansions and inner expansions
  9. Proof of Proposition \ref{['thm:approx_sol_scalarEQ']}
  10. Spectral analysis of the approximate solutions
  11. Spectral analysis of the approximate solutions
  12. The principal eigenvalue of $\mathcal{L}^{\varepsilon}_k$
  13. Proofs of Lemmas \ref{['lem:exp_decay_nearbc']} and \ref{['lem:exp_decay_on_Ball']}
  14. Proof of Theorem \ref{['thm:mathscr_L_k^vep']}
  15. Proof of Lemma \ref{['lem:mathscr L_k^vep inver_bdd']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that conditions (A1)-(A3) hold. Then, for each given there exists a constant $\bar{\varepsilon} = \bar{\varepsilon}(M, D_0)>0$ such that system eqn:MCRD-ss admits a family of spherically symmetric classical solutions $(u^{\varepsilon, D}(r), v^{\varepsilon, D}(r))$ defined for $\varepsilon\in(0, \bar{\varepsilon}]$ and $D\geq D_0$ and satisfying the m Moreover, these solutions satisfy the

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Remark 2.4
  • proof : Proof of Lemma \ref{['lem:L^0-solvabilty']}
  • Proposition 2.5
  • Proposition 2.6
  • ...and 42 more