Asymptotic spreading for general heterogeneous Fisher-KPP type equations

Abstract : In this article, we establish spreading properties for heterogeneous Fisher-KPP reaction-diffusion equations for initial data with compact support, where the nonlinearity admits 0 as an unstable steady state and 1 as a globally attractive one. Here, the coefficients are only assumed to be uniformly elliptic, continuous and bounded in (t, x). We construct two non-empty star-shaped compact sets such that for all compact set K in the interior of the first one (resp. all closed set F in the complementary of the second one), one has lim t→+∞ sup x∈tK |u(t, x) − 1| = 0 (resp. lim t→+∞ sup x∈tF |u(t, x)| = 0). The characterization of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. It gives in particular an exact asymptotic speed of propagation for almost periodic, asymptotically almost periodic and radially periodic equations (where the two sets are identical) and explicit bounds on the location of the transition between 0 and 1 in spatially homogeneous equations. In dimension N , if the coefficients converge in radial segments, then the two sets are identical and this set is characterized using some geometric optics minimization problem, which may give rise to non-convex expansion sets.
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Submitted on : Monday, August 5, 2019 - 4:33:38 PM
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  • HAL Id : hal-01171334, version 4

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Henri Berestycki, Grégoire Nadin. Asymptotic spreading for general heterogeneous Fisher-KPP type equations. Memoirs of the American Mathematical Society, American Mathematical Society, In press. ⟨hal-01171334v4⟩

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