In Part II of this series of papers, we consider an initial‐boundary value problem for the Kolmogorov–Petrovskii–Piscounov (KPP)‐type equation with a discontinuous cut‐off in the reaction function at concentration urn:x-wiley:00222526:media:sapm12352:sapm12352-math-0001. For fixed cut‐off value urn:x-wiley:00222526:media:sapm12352:sapm12352-math-0002, we apply the method of matched asymptotic coordinate expansions to obtain the complete large‐time asymptotic form of the solution, which exhibits the formation of a permanent form traveling wave (PTW) structure. In particular, this approach allows the correction to the wave speed and the rate of convergence of the solution onto the PTW to be determined via a detailed analysis of the asymptotic structures in small time and, subsequently, in large space. The asymptotic results are confirmed against numerical results obtained for the particular case of a cut‐off Fisher reaction function.
|Journal||Studies in Applied Mathematics|
|Early online date||16 Dec 2020|
|Publication status||Published - 1 Feb 2021|
- matched asymptotic expansions
- permanent form traveling waves
- reaction–diffusion equations