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

Abstract

The large time behavior of solutions to the following generalized Burgers-Fisher-KPP equation $${\partial }_{t}u=u_{xx}+k(u^n{)}_x+u^p-u^q,(x,t)\in \mathbb{R}\times (0,\infty ),$$ with $n\ge 2$, $p>q\ge 1$ and $k\in \mathbb{R}$, is considered in this work. Denoting by $H(x,t)$, respectively $\tilde{H}(x,t)$ 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 $\tilde{c}=kn+2\sqrt{p-q}$, are identified such that $H(x,t)$, respectively $\tilde{H}(x,t)$ 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 $\tilde{H}(x,t)\to 0$ 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 $k^{\ast }(n,p,q)$ such that $\overline{c}>0$ if $k>k^{\ast }(n,p,q)$ and $\overline{c}<0$ if $k<k^{\ast }(n,p,q)$. 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 $k^{\ast }(n,p,q)$ are also given.