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)∂tu−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u) with compactly supported initial conditions at t=0t=0. Here, A,q,fA,q,f have a general dependence in t∈R+t∈R+ and x∈RNx∈RN. We establish properties of families of propagation sets   which are defined as families of subsets (St)t⩾0(St)t⩾0 of RNRN such that lim inft→+∞{infx∈Stu(t,x)}>0lim inft→+∞{infx∈Stu(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({ξ(t)})t⩾0, where ξ(t)∈RNξ(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)=+∞limt→+∞r(t)=+∞. This second property is called here complete spreading  . Furthermore, in the case q≡0q≡0 and inf(t,x)∈R+×RNfu′(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)r(t) can be chosen so that lim inft→+∞r(t)/t>0lim inft→+∞r(t)/t>0. In the general case, we also show the existence of an explicit upper bound C>0C>0 such that lim supt→+∞r(t)/t0ε>0, any family of propagation sets (St)t⩾0(St)t⩾0 has to satisfy St⊂{x∈RN,|x|⩽εt} for large t. In connection with spreading properties, we derive some new uniqueness 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.