Asymptotic spreading in heterogeneous diffusive excitable media

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type: partial derivative(t)u - del center dot (A(t, x)del u) + q(t, x) center dot del u = f(t, x, u) with compactly supported initial conditions at t = 0. Here, A, q, f have a general dependence in t is an element of R+ and x is an element of R-N. We establish properties of families of propagation sets which are defined as families of subsets (S-t)(t >= 0) of R-N such that lim inf(t ->+infinity){inf(x is an element of 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 ({xi(t)})(t >= 0), where xi(t) is an element of R-N, forms a family of propagation sets, or (2) one can find such a family with S-t superset of {x is an element of R-N, |x| <= r(t)} and lim(t ->+infinity)r(t) = +infinity. This second property is called here coniplete spreading. Furthermore, in the case q equivalent to 0 and inf((t, x)is an element of R+ x 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 inf(t ->+infinity)r(t)/t > 0. In the general case, we also show the existence of an explicit upper bound C > 0 such that lim sup(t ->+infinity)r(t)/t < C. On the other hand, we provide explicit examples of reactiondiffusion equations such that for an arbitrary epsilon > 0, any family of propagation sets (S-t)(t >= 0) has to satisfy S-t subset of {x is an element of R-N, |x| <= epsilon t} 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. (C) 2008 Elsevier Inc. All rights reserved.