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| In this article, we establish spreading properties for heterogeneous Fisher-KPP reaction-diffusion equations of the type:
\begin{equation} \partial_{t} u - \Delta u =c(t,x) u (1-u)
\end{equation}
for initial data with compact support, where the growth rate $c$ is only assumed to be uniformly continuous and bounded in $(t,x)$, without any specific assumption such as periodicity.
Our aim is to localize the transition between the stable steady state $1$ and the unstable one $0$
in any direction $|e|=1$. Namely, we construct two speeds such that
$$\lim_{t\to +\infty} u(t,wte)= 0 \hbox{ if } w>\overline{w}_e, \quad 1 \hbox{ if } w\in (0,\underline{w}_e).$$
The characterization of these speeds involve two new notions of
generalized principal eigenvalues for linear operators in unbounded domains.
It gives in particular an exact asymptotic speed of propagation for almost periodic, asymptotically almost periodic and radially periodic equations $\underline{w}_e=\overline{w}_e$ and explicit bounds on the location of the transition between $0$ and $1$ in spatially homogeneous equations.
In dimension $N$, if the coefficients converge in radial segments, then $\underline{w}_e=\overline{w}_e$.
and this set is characterized using some geometric optics minimization problem, which may give rise to non-convex expansion sets. |
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