Asymptotic spreading in heterogeneous diffusive excitable media

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction–diffusion equations of the type: ∂tu−∇ · A(t, x)∇u +q(t,x) · ∇u = f (t,x,u) with compactly supported initial conditions at t = 0. Here, A,q,f have a general dependence in t ∈ R+ and x ∈ RN. We establish properties of families of propagation sets which are defined as families of subsets (St )t 0 of RN such that lim inft→+∞{infx∈St u(t, x)} > 0. The aim is to characterize such families as sharply as possible. In particular, we give some conditions under which: (1) a given path ({ξ(t)})t 0, where ξ(t) ∈ RN, forms a family of propagation sets, or (2) one can find such a family with St ⊃ {x ∈ RN, |x| r(t)} and limt→+∞r(t) = +∞. This second property is called here complete spreading. Furthermore, in the case q ≡ 0 and inf(t,x)∈R+×RN f u(t, x, 0) > 0, as well as under some more general assumptions, we show that there is a positive spreading speed, that is, r(t) can be chosen so that lim inft→+∞r(t)/t > 0. In the general case, we also show the existence of an explicit upper bound C >0 such that lim supt→+∞r(t)/t < C. On the other hand, we provide explicit examples of reaction– diffusion equations such that for an arbitrary ε > 0, any family of propagation sets (St )t 0 has to satisfy St ⊂ {x ∈ RN, |x| εt} for large t . In connection with spreading properties, we derive some new unique-ness results for the entire solutions of this type of equations. Lastly, in the case of space–time periodic media, we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero.