Abstract
In this paper, non-planar two-dimensional travelling fronts connecting an unstable one-dimensional periodic limiting state to a constant stable state are constructed for some reaction-diffusion equations with bistable nonlinearities. The minimal speeds are characterized in terms of the spatial period of the unstable limiting state. The limits of the minimal speeds and of the travelling fronts as the period converges to a critical minimal value or to infinity are analyzed. The fronts converge to flat fronts or to some curved fronts connecting an unstable ground state to a constant stable state.
Highlights
Introduction and main resultsThis paper is concerned with special time-global solutions v(t, x, y) of the reaction-diffusion equation ∂v − ∆v = f (v) ∂t (1.1)set in the two-dimensional space R2 = (x, y), x ∈ R, y ∈ R
Remark 1.5 Throughout the paper, the solutions of (1.2) or (1.9) given in Theorems 1.1 and 1.4 are two-dimensional y-connections between an unstable limiting profile φ and the stable constant state 1. Each of these solutions uc,L or uc,∞ is expected to be stable for the Cauchy problem (1.1), at least if the initial condition v0 is above max φL or max φ∞ as y → +∞ and if the initial perturbation v0 − uc,L or v0 − uc,∞ decays to 0 as y → −∞
The methods are different: the paper [21] is based on the direct construction of suitable sub- and super-solutions for problem (1.9), while our construction is based on approximated problems which are L-periodic in x and whose properties are analyzed in this paper
Summary
This paper is concerned with special time-global solutions v(t, x, y) of the reaction-diffusion equation. Remark 1.5 Throughout the paper, the solutions of (1.2) or (1.9) given in Theorems 1.1 and 1.4 are two-dimensional y-connections between an unstable limiting profile φ and the stable constant state 1. Each of these solutions uc,L or uc,∞ is expected to be stable for the Cauchy problem (1.1), at least if the initial condition v0 is above max φL or max φ∞ as y → +∞ and if the initial perturbation v0 − uc,L or v0 − uc,∞ decays to 0 as y → −∞. The methods are different: the paper [21] is based on the direct construction of suitable sub- and super-solutions for problem (1.9), while our construction is based on approximated problems which are L-periodic in x and whose properties are analyzed in this paper
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