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We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible flows. We prove the existence of and determine the limits c * (A)/A and D(A)/A 2 as A → ∞, where c * (A) is the minimal front speed and D(A) the effective diffusivity.
We obtain a criterion for pulsating front speed-up by general periodic incompressible flows in two dimensions and in the presence of KPP nonlinearities. We achieve this by showing that the ratio of the minimal front speed and the effective diffusivity of the flow is bounded away from zero and infinity by constants independent of the flow. We also study… (More)
We consider the reaction-diffusion equation T t = T xx + f (T) on R with T 0 (x) ≡ χ [−L,L] (x) and f (0) = f (1) = 0. In 1964 Kanel' proved that if f is an ignition non-linearity, then T → 0 as t → ∞ when L < L 0 , and T → 1 when L > L 1. We answer the open question of relation of L 0 and L 1 by showing that L 0 = L 1. We also determine the large time… (More)
We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion… (More)
We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomo-geneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic… (More)
We use a new method in the study of Fisher-KPP reaction-diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction-diffusion equations in several spatial dimensions. Our method is based on the construction of sub-and… (More)
We consider the advection-diffusion equation φ t + Au · ∇φ = ∆φ, φ(0, x) = φ 0 (x) on R 2 , with u a periodic incompressible flow and A 1 its amplitude. We provide a sharp characterization of all u that optimally enhance dissipation in the sense that for any initial datum φ 0 ∈ L p (R 2), p < ∞, and any τ > 0, Our characterization is expressed in terms of… (More)
We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove… (More)
We prove that there are solutions to the Euler equation on the torus with C 1,α vorticity and smooth except at one point such that the vorticity gradient grows in L ∞ at least exponentially as t → ∞. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev andŠverák .