We study nonlinear diffusion problems of the form ut = uxx + f (u) with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. For monostable, bistable, and combustion types of nonlinearities, Du and Lou ["Spreading and vanishing in nonlinear diffusion problems with free boundaries," J. Eur. Math. Soc. (JEMS), to appear] obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between spreading (limt→∞ u(t, x) = 1) and vanishing (limt→∞ u(t, x) = 0). They also determined the asymptotic spreading speed of the fronts by making use of semiwaves when spreading happens. In this paper, we give a much sharper estimate for the spreading speed of the fronts than that in the above-mentioned work of Du and Lou, and we describe how the solution approaches the semiwave when spreading happens.

We consider Fisher-KPP equation with advection: u t = u xx − βu x + f (u) for x ∈ (g(t), h(t)), where g(t) and h(t) are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient −β on the long time behavior of the solutions. We find two parameters c 0 and β * with β * > c 0 > 0 which play key roles in the dynamics, here c 0 is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data {σφ} σ>0 (where φ is some compactly supported positive function), we show that: (1) in case β ∈ (0, c 0 ), there exists σ * 0 such that spreading happens when σ > σ * (i.e., u(t, · ; σφ) → 1 locally uniformly in R) and vanishing happens when σ ∈ (0, σ * ] (i.e., [g(t), h(t)] remains bounded and u(t, · ; σφ) → 0 uniformly in [g(t), h(t)]); (2) in case β ∈ (c 0 , β * ), there exists σ * > 0 such that virtual spreading happens when σ > σ * (i.e., u(t, · ; σφ) → 0 locally uniformly in [g(t), ∞) and u(t, · + ct; σφ) → 1 locally uniformly in R for some c > β − c 0 ), vanishing happens when σ ∈ (0, σ * ), and in the transition case σ = σ * , u(t, · + o(t); σφ) → V * (· − (β − c 0 )t) uniformly, the latter is a traveling wave with a "big head" near the free ✩ 1715 boundary x = (β − c 0 )t and with an infinite long "tail" on the left; (3) in case β = c 0 , there exists σ * > 0 such that virtual spreading happens when σ > σ * and u(t, · ; σφ) → 0 uniformly in [g(t), h(t)] when σ ∈ (0, σ * ]; (4) in case β β * , vanishing happens for any solution.

Abstract. This paper continues the investigation of Du and Lou [5], where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form ut = uxx + f (u) for x over a varying interval (g(t), h(t)) was examined. Here x = g(t) and x = h(t) are free boundaries evolving according to g, and u(t, g(t)) = u(t, h(t)) = 0. We answer several intriguing questions left open in [5]. First we prove the conjectured convergence result in [5] for the general case that f is C 1 and f (0) = 0. Second, for bistable and combustion types of f , we determine the asymptotic propagation speed of h(t) and g(t) in the transition case. More presicely, we show that when the transition case happens, for bistable type of f there exists a uniquely determined c1 > 0 such that limt→∞ h(t)/ ln t = limt→∞ −g(t)/ ln t = c1, and for combustion type of f , there exists a uniquely determined c2 > 0 such that limt→∞ h(t)/ √ t = limt→∞ −g(t)/ √ t = c2. Our approach is based on the zero number arguments of Matano and Angenent, and on the construction of delicate upper and lower solutions.

Abstract-An accurate and efficient direct optimization technique for the design of contoured beam reflectarrays is presented. It is based on the spectral domain method of moments assuming local periodicity and minimax optimization. Contrary to the conventional phase-only optimization techniques, the geometrical parameters of the array elements are directly optimized to fulfill the contoured beam requirements, thus maintaining a direct relation between optimization goals and optimization variables, and hence resulting in more optimal designs. Both co-and cross-polar radiation patterns of the reflectarray can be optimized for multiple frequencies, polarizations, and feed illuminations. Several contoured beam reflectarrays, that radiate a high-gain beam on a European coverage, have been designed and compared to similar designs obtained using the phase-only optimization technique. The comparisons show that the designs obtained using the proposed direct optimization technique are superior in performance, both for multi-frequency and dual-polarization designs. A reflectarray breadboard has been manufactured and measured at the DTU-ESA Spherical Near-Field Antenna Test Facility to validate the proposed technique. An excellent agreement of the simulated and measured patterns is obtained.

Abstract. We determine the asymptotic spreading speed of an invasive species, which invades the territory of a native competitor, governed by a diffusive competition model with a free boundary in a spherically symmetric setting. This free boundary problem was studied recently in [8], but only rough bounds of the spreading speed was obtained there. We show in this paper that there exists an asymptotic spreading speed, which is determined by a certain traveling wave type system of one space dimension, called a semi-wave. This appears to be the first result that gives the precise asymptotic spreading speed for a two species system with free boundaries.