Front propagation into unstable states for periodic monotone reaction-diffusion systems

Liangliang Deng; Arnaud Ducrot; Quentin Griette

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arXiv · 2503.11369

Front propagation into unstable states for periodic monotone reaction-diffusion systems

Liangliang Deng·Arnaud Ducrot·Quentin Griette

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Abstract

In this paper we study the invasion fronts of spatially periodic monotone reaction-diffusion systems in a multi-dimensional setting. We study the pulsating traveling waves that connect the trivial equilibrium, for which all components of the state variable are identically equal to zero, to a uniformly persistent stationary state, for which all components are uniformly positive. When the trivial equilibrium is linearly unstable, we show that all pulsating traveling waves have a speed that is greater than the speed of the linearized system at the equilibrium, in any given direction. If moreover the nonlinearity is sublinear, then we can construct a pulsating traveling wave that travels at any super-linear speed in any given direction (i.e. the minimal speed is linearly determined). We also show that pulsating traveling waves are monotonic in time as soon as the nonlinearity is sub-homogeneous. Beyond these general qualitative properties, the main focus of the paper is to derive sufficient conditions for the existence and nonexistence of pulsating waves propagating in any given direction. Our proof of the existence part relies upon a new level of understanding of the multi-dimensional pulsating waves observed from a direction-dependent coordinate system.

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Liangliang Deng, Arnaud Ducrot, Quentin Griette. 2025-03-14. Front propagation into unstable states for periodic monotone reaction-diffusion systems. https://arxiv.org/abs/2503.11369

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