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

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.

💡 Research Summary

The paper addresses the rigorous existence of stationary patterns with sharp internal interfaces in mass‑conserving reaction‑diffusion (MCRD) systems that feature a bistable reaction term. While numerical studies and formal asymptotics have long suggested that such systems can support “transition‑layer” solutions—regions where the concentration of one component jumps abruptly across a thin layer—mathematical proofs have been limited to one‑dimensional domains. This work extends the theory to arbitrary spatial dimension N, focusing on radially symmetric solutions on an N‑dimensional ball.

The authors begin with a general two‑component MCRD system
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