  We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP)
fronts in heterogeneous shifting habitats, with any number of shifting speeds,
by further developing the method based on the theory of viscosity solutions of
Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion
equation and integro-differential equations with a distributed time-delay. The
latter leads to a class of limiting equations of Hamilton-Jacobi-type depending
on the variable $x/t$ and in which the time and space derivatives are coupled
together. We will first establish uniqueness results for these Hamilton-Jacobi
equations using elementary arguments, and then characterize the spreading speed
in terms of a reduced equation on a one-dimensional domain in the variable
$s=x/t$. In terms of the standard Fisher-KPP equation, our results leads to a
new class of "asymptotically homogeneous" environments which share the same
spreading speed with the corresponding homogeneous environments.

Analysis of PDEs