Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity

The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$ u_t(t,x)=(\mathcal{A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quad x\in\mathcal{H}, $$ where $\mathcal{H}=\RR^N$ or $\ZZ^N$, $\mathcal{A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\mathcal{H}=\RR^N$ and is a discrete dispersal operator in the case $\mathcal{H}=\ZZ^N$, and $f$ is periodic in $t$, asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to 0 as $\|x\|\to\infty$ for some time and space periodic function $f_0(t,x,u)$), and is of KPP type in $u$. It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if $u\equiv 0$ is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction.