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The evolution of traveling waves in a KPP reaction-diffusion model with cut-off reaction rate. II. Evolution of traveling waves. (English) Zbl 1467.35089

Summary: 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 \(u=u_c\). For fixed cut-off value \(u \in (0,1)\), 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.
For Part I, see [the authors, ibid. 146, No. 2, 301–329 (2021; Zbl 1467.35088)].

MSC:

35C07 Traveling wave solutions
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K57 Reaction-diffusion equations

Citations:

Zbl 1467.35088
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