Large time behavior and transition from vanishing to spreading regimes for the generalized Burgers-Fisher-KPP equation

TL;DR

This paper analyzes the long-term behavior of solutions to a generalized Burgers-Fisher-KPP equation, identifying critical velocities and the influence of convection on solution limits, with results applicable to various initial conditions.

Contribution

It introduces the concept of an anomalous critical velocity and establishes the existence of a threshold coefficient dictating solution convergence, extending understanding of the equation's dynamics.

Findings

Solutions approach traveling waves at critical velocities.

The sign of the critical velocity depends on parameters and convection strength.

Convection term significantly influences whether solutions tend to zero or one.

Abstract

The large time behavior of solutions to the following generalized Burgers-Fisher-KPP equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k\in\mathbb{R}$, is considered in this work. Denoting by $H(x,t)$, respectively $\widetilde{H}(x,t)$ the solutions having as initial condition the Heaviside, respectively the ``anti-Heaviside" functions $$ H_0(x)=\begin{cases} 0, & \mbox{if } x<0 1, & \mbox{if } x\geq0. \end{cases}, \quad \widetilde{H}_0(x)=1-H_0(x), $$ critical velocities $\overline{c}$, respectively $\widetilde{c}=kn+2\sqrt{p-q}$, are identified such that $H(x,t)$, respectively $\widetilde{H}(x,t)$ approach the unique traveling wave solution of the equation with these critical velocities as $t\to\infty$. The critical velocity $\overline{c}$ is \emph{anomalous}, that is, it cannot be made explicit by an algebraic expression. Assuming for simplicity $k>0$, a remarkable fact is that, while $\widetilde{H}(x,t)\to0$ as $t\to\infty$ uniformly on compact subsets of $\mathbb{R}$, the Heaviside solution $H$ might tend either to zero or to one as $t\to\infty$, depending on the sign of the critical velocity $\overline{c}$. This sign vary with respect to the exponents $n$, $p$, $q$ and the coefficient $k$ and, in fact, we prove that given $p$, $q$, $n$, there exists a critical coefficient $k^*(n,p,q)$ such that $\overline{c}>0$ if $k>k^*(n,p,q)$ and $\overline{c}<0$ if $k<k^*(n,p,q)$. The convergence to either zero or one reflects the sharp influence of the convection term, since in the absence of it (that is, $k=0$), $H(x,t)$ would always tend to zero as $t\to\infty$. The results include more general initial conditions than the Heaviside-type functions, and sharp estimates of the threshold coefficient $k^*(n,p,q)$ are also given.