Spreading speeds for one-dimensional monostable reaction-diffusion equations

By Henri Berestycki, Grégoire Nadin. 2016

54 citations recorded by the source corpus.

We establish in this article spreading properties for the solutions of equations of the type $\partial$ t u -- a(x)$\partial$ xx u -- q(x)$\partial$ x u = f (x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable KPP type between two steady states 0 and 1 and the initial datum is compactly sup-ported. Using homogenization techniques, we construct two speeds w $\le$ w such that lim t$\rightarrow$+$\infty$ sup 0$\le$x$\le$wt |u(t, x)--1| = 0 for all w $\in$ (0, w) and lim t$\rightarrow$+$\infty$ sup x$\ge$wt |u(t, x)| = 0 for all w \textgreater{} w. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particu-lar, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where w = w).

Subfields: Applied Mathematics, Computational Theory and Mathematics, Public Health, Environmental and Occupational Health

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In Applied Mathematics

In Computational Theory and Mathematics

In Public Health, Environmental and Occupational Health

Citations in the corpus

On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “Lately, the search of Liouville type results for semilinear elliptic equations in unbounded domains [10, 56] and the characterisation of spreading speed [7, 48] have stimulated the studies of the properties of λ1(E) and several other notions of principal eigenvalue have emerged.”