Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations

Source abstract

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: ∂ t u − ∑ i , j = 1 N a i , j ( t , x ) ∂ i j u − ∑ i = 1 N q i ( t , x ) ∂ i u = f ( t , x , u ) . $$\begin{equation*} \partial _{t} u - \sum _{i,j=1}^N a_{i,j}(t,x)\partial _{ij}u-\sum _{i=1}^N q_i(t,x)\partial _i u=f(t,x,u). \end{equation*}$$ These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity f f is of Fisher-KPP type, and admits 0 0 as an unstable steady state and 1 1 as a globally attractive one (or, more generally, admits entire solutions p ± ( t , x ) p^\pm (t,x) , where p − p^- is unstable and p + p^+ is globally attractive). Here, the coefficients a i , j , q i , f a_{i,j}, q_i, f are only assumed to be uniformly elliptic, continuous and bounded in ( t , x ) (t,x) . To describe the spreading dynamics, we construct two non-empty star-shaped compact sets S _ ⊂ S ¯ ⊂ R N \underline {\mathcal {S}}\subset \overline {\mathcal {S}} \subset \mathbb {R}^N such that for all compact set K ⊂ i n t ( S _ ) K\subset \mathrm {int}(\underline {\mathcal {S}}) (resp. all closed set F ⊂ R N ∖ S ¯ F\subset \mathbb {R}^N\backslash \overline {\mathcal {S}} ), one has lim t → + ∞ sup x ∈ t K | u ( t , x ) − 1 | = 0 \lim _{t\to +\infty } \sup _{x\in tK} |u(t,x)-1| = 0 (resp. lim t → + ∞ sup x ∈ t F | u ( t , x ) | = 0 \lim _{t\to +\infty } \sup _{x\in tF} |u(t,x)| =0 ). The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {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 N , if the coefficients converge in radial segments, again we show that S ¯ = S _ \overline {\mathcal {S}}=\underline {\mathcal {S}} and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.