Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations:partial derivative tu- Sigma(N) (i,j =1) ai,j(t, x) partial derivative iju - Sigma(N) (i=1) qi(t, x) partial derivative iu = f (t, x, u).These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity f is of Fisher-KPP type, and admits 0 as an unstable steady state and 1 as a globally attractive one (or, more generally, admits entire solutions p +/-(t, x), where p- is unstable and p+ is globally attractive). Here, the coefficients ai,j, qi, f are only assumed to be uniformly elliptic, continuous and bounded in (t, x). To describe the spreading dynamics, we construct two non-empty star-shaped compact sets S(sic) subset of(s) over bar subset of R-N such that for all compact set K subset of int(S(sic)) (resp. all closed set F subset of R-N\(s) over bar), one has limt ->+infinity sup(x is an element of tK) |u(t, x) - 1| = 0 (resp. limt ->+infinity sup(x is an element of tF) |u(t, x)| = 0). The characterizations of these sets involve two new notions of generalized princi-pal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that S = S and to establish an exact asymptotic speed of propagation in various frameworks. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension N, if the coefficients converge in radial segments, again we show that S = S and this set is characterized using some geo-metric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.