Long time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(theta_tomega)u(1-u),quad xinmathbb{R} $$ where $omegainOmega$, $(Omega, mathcal{F},mathbb{P})$ is a given probability space, $theta_t$ is an ergodic metric dynamical system on $Omega$, and $a(omega)>0$ for every $omegainOmega$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),quad xinmathbb{R}$$ where $a_0(t)$ is a positive locally Holder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $a(omega)$, we show that the constant solution $u=1$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $[2sqrt{underline a}, 2sqrt{bar a}]$, where $underline{a}$ and $bar a$ are the least and the greatest means of $a(cdot)$, respectively, and hence the spreading speed interval is linearly determinant. It is shown that the solution of (1) with the initial function which is bounded away from $0$ for $xll -1$ and is $0$ for $xgg 1$ propagates at the speed $2sqrt {hat a}$, where $hat a$ is the average of $a(cdot)$. Under some assumption on $a_0(cdot)$, we also show that the constant solution $u=1$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $a(omega)$ and $a_0(t)$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher-KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).