We consider a spatially homogeneous system of reaction-diffusion equation defined on the interval (-∞, ∞) of the one-dimensional spatial variable x. It is known that this equation has a one-parameter family of periodic travelling wave solutions Ψ(x + ct; c) if this equation has a spatially homogeneous periodic solution φ(t). The spatial period L(c) of the travelling wave solution satisfies L(c) c → T if c → +∞, where c is the propagation speed and T is the period of φ(t). We prove that, in the case c > 0 is sufficiently large, Ψ(x + ct; c) is unstable if φ(t) is "strongly unstable" and Ψ(x + ct; c) is "marginally stable" if φ(t) is "strongly stable." If the equation is defined on a finite interval [0, l] of the variable x with the periodic boundary conditions, we can obtain a more precise result regarding the stability of Ψ(x + c ̂t; c ̂), where c ̂ > 0 is a speed which satisfies l = mL( c ̂) for some positive integer m. We prove that this solution is asymptotically stable in the sense of waveform stability if c ̂ > 0 is sufficiently large and if φ(t) is "strongly stable.". © 1981.
Maginu, K. (1981). Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems. Journal of Differential Equations, 39(1), 73–99. https://doi.org/10.1016/0022-0396(81)90084-X