Asymptotic dependence of oscillatory traveling waves on the viscosity coefficient

Characterize precisely how the asymptotic profile of traveling-wave solutions of the pseudo-parabolic Fisher–KPP equation $u_t- au u_{xxt}=D u_{xx}+u(1-u)$ depends on the viscosity coefficient $ au$ when $ au>D$, including the precise influence of $ au$ on the amplitude and frequency of the oscillations near the equilibrium $u=1$.

Background

The paper proves that the ratio $au/D$ determines a transition in the traveling-wave structure: waves are monotone for $0< au/D\leq1$ and oscillatory near the equilibrium $u=1$ for $au/D>1$. Numerical simulations further indicate that, for $au>D$, both the amplitude and frequency of the oscillations increase as $au$ increases.

Although the existence and qualitative non-monotonicity of the waves are established, the authors do not derive a precise mathematical description of the asymptotic oscillatory profile or quantify its dependence on $au$. This unresolved problem concerns the detailed amplitude and frequency behavior of the oscillatory tails.