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Journals and Conferences
(1.2) Tt = Txx + f(T ) with x ∈ R. This equation has been extensively studied in mathematical, physical and other literature, starting with the pioneering works of Fisher  and Kolmogorov, Petrovskii, Piskunov . In these papers (1.2) was used to describe the propagation of advantageous genes in a population. The main object of study in these and many… (More)
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 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 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 reactiondiffusion equations in several spatial dimensions. Our method is based on the construction of suband… (More)
We prove that there are solutions to the Euler equation on the torus with C 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 and Šverák .
Abstract. 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… (More)
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 as A → ∞, where c∗(A) is the minimal front speed and D(A) the effective diffusivity.
We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous 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 study a reaction-diffusion equation in the cylinder Ω = R×T, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of T larger than three as T → 0 and the initial datum is small, then the flame is extinguished — the solution quenches. If, on… (More)
We consider the advection-diffusion equation φt +Au · ∇φ = ∆φ, φ(0, x) = φ0(x) on R, 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 φ0 ∈ L(R), p <∞, and any τ > 0, ‖φ(·, τ)‖∞ → 0 as A→∞. Our characterization is expressed in… (More)