arXiv subjects

Finn Münnich

Publications and source records attributed to Finn Münnich.

3 recordsLinked to original sources

arXiv2026

Multiple diffusion scales and diffusion-driven instability: Emergence of near- and far-from-equilibrium patterns

This paper investigates pattern formation in reaction--diffusion systems with both diffusive and nondiffusive components, providing necessary and sufficient conditions for diffusion-driven instability (DDI) and establishing the existence of far-from-equilibrium patterns. While previous work has linked DDI to instability in the purely nondiffusive subsystem -- thereby destabilizing all regular Turing patterns -- we show that DDI can also arise from subsystems involving nondiffusive and slow-diffusive components using three different spatial scales. This leads to simple sufficient conditions for DDI in systems with arbitrary numbers of components. Moreover, we fully classify all possible sources of DDI in the case of two diffusive and one nondiffusive component. Further, we prove the existence of far-from-equilibrium patterns exhibiting branch-switching and discontinuities in the nondiffusive components, which cannot occur in classical reaction--diffusion equations. We illustrate our results with a receptor-based model supported by numerical bifurcation analysis and simulations. These findings extend the theoretical foundations of pattern formation, demonstrating how coupling between diffusive and nondiffusive dynamics can generate patterns beyond the reach of the classical reaction--diffusion framework.

math.AP↗

arXiv2026

Stability and bifurcation analysis in a mechanochemical model of pattern formation

We analyze the stability and bifurcation structure of steady states in a mechanochemical model of pattern formation in regenerating tissue spheroids. The model couples morphogen dynamics with tissue mechanics via a positive feedback loop: mechanical stretching enhances morphogen production, while morphogen concentration modulates tissue elasticity. Global strain conservation implements a nonlocal inhibitory effect, realizing a mechanochemical variant of the local activation--long-range inhibition mechanism. For exponential elasticity-morphogen coupling, the system admits a variational formulation. We prove existence of nonconstant steady states for small diffusion and uniqueness of the homogeneous state for large diffusion. Linear stability analysis shows that only unimodal patterns are stable, while multimodal solutions are unstable. Bifurcation analysis reveals subcritical and supercritical pitchforks, with fold bifurcations generating bistable regimes. Our results demonstrate that mechanochemical feedback provides a robust mechanism for single-peaked pattern formation without requiring a second diffusible inhibitor.

math.AP↗

arXiv2023

Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems

Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially low-regular stationary solutions. This work provides a comprehensive theory of asymptotic stability of bounded, discontinuous or continuous, stationary solutions of reaction-diffusion-ODE systems. We characterize the spectrum of the linearized operator and relate its spectral properties to the corresponding semigroup properties. Considering the function spaces $L^\infty(Ω)^{m+k}, L^\infty(Ω)^m \times C(\overlineΩ)^k$ and $C(\overlineΩ)^{m+k}$, we establish a sign condition on the spectral bound of the linearized operator, which implies nonlinear stability or instability of the stationary pattern.

math.AP↗