Sharp estimates on minimum travelling wave speed of reaction diffusion systems modelling autocatalysis

This article studies propagating wave fronts in an isothermal chemical reaction A + 2B - \u3e 3B involving two chemical species, a reactant A and an autocatalyst B, whose diffusion coefficients, D-A and D-B, are unequal due to different molecular weights and/or sizes. Explicit bounds v(*) and v* that depend on D-B/D-A are derived such that there is a unique travelling wave of every speed v \u3e = v* and there does not exist any travelling wave of speed v \u3c v*. New to the literature, it is shown that v(*) proportional to v* proportional to D-B/D-A when D-B = v(min). Estimates on v(min) significantly improve those of early works. The framework is built upon general isothermal autocatalytic chemical reactions A + nB - \u3e (n + 1)B of arbitrary order n \u3e = 1

Mass conserved Allen-Cahn equation and volume preserving mean curvature flow

We consider a mass conserved Allen-Cahn equation u_t=\Delta u+ \e^{-2} (f(u)-\e\lambda(t)) in a bounded domain with no flux boundary condition, where \e\lambda(t) is the average of $f(u(\cdot,t))$ and $-f$ is the derivative of a double equal well potential. Given a smooth hypersurface $\gamma_0$ contained in the domain, we show that the solution u^\e with appropriate initial data approaches, as \e\searrow0, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from $\gamma_0$

Acoustic scattering by an inhomogeneous layer on a rigid plate

The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data

Propagation of Local Disturbances in Reaction Diffusion Systems Modeling Quadratic Autocatalysis

This article studies the propagation of initial disturbance in a quadratic autocatalytic chemical reaction in one-dimensional slab geometry, where two chemical species A, called the reactant, and B, called the autocatalyst, are involved in the simple scheme A + B - \u3e 2B. Experiments demonstrate that chemical systems for which quadratic or cubic catalysis forms a key step can support propagating chemical wavefronts. When the autocatalyst is introduced locally into an expanse of the reactant, which is initially at uniform concentration, the developing reaction is often observed to generate two wavefronts, which propagate outward from the initial reaction zone. We show rigorously that with such an initial setting the spatial region is divided into three regions by the two wavefronts. In the middle expanding region, the reactant is almost consumed so that A approximate to 0, whereas in the other two regions there is basically no reaction so that B approximate to 0. Most of the chemical reaction takes place near the wavefronts. The detailed characterization of the concentrations is given for each of the three zones

Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers

We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be sufficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations

Analysis of an Optimal Stopping Problem Arising from Hedge Fund Investing

The final publication is available at Elsevier via https://doi.org/10.1016/j.jmaa.2019.123559. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We analyze the optimal withdrawal time for an investor in a hedge fund with a first-loss or shared-loss fee structure, given as the solution of an optimal stopping problem on the fund's assets with a piecewise linear payoff function. Assuming that the underlying follows a geometric Brownian motion, we present a complete solution of the problem in the infinite horizon case, showing that the continuation region is a finite interval, and that the smooth-fit condition may fail to hold at one of the endpoints. In the finite horizon case, we show the existence of a pair of optimal exercise boundaries and analyze their properties, including smoothness and convexity.NSERC, RGPIN-2017-04220