Dynamical interface transition in ramified media with diffusion

We consider interaction problems in ramified spaces of a rather general type with a distinguished interface transition in form of a dynamical Kirchhoff condition. These conditions stem from applications and modeling in biology and physics, see for instance [8,21]. In the present paper our principal concern is to derive existence results in the linear and semilinear case and some qualitative principles similar to the classical ones for initial boundary value problems on domains. Moreover, the influence of the dynamical interface condition on the solution is studied in the autonomous reaction-diffusin case.     

Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogeneous media

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through interfaces, which is mediated by a suitable Laplace-Beltrami type equation.     

Positive solutions of a diffusive prey–predator model in a heterogeneous environment

In this paper, we investigate a diffusive prey–predator model in a spatially degenerate heterogeneous environment. We are concerned with the positive solutions of the model, and obtain some results for the existence and non-existence of positive solutions. Moreover, the multiplicity, stability and asymptotical behaviors of positive solutions with respect to the parameters are also studied.     

Spreading and vanishing in a diffusive intraguild predation model with intraspecific competition and free boundary

This paper is concerned with the spreading and vanishing phenomena in a diffusive intraguild (IG) predation model with intraspecific competition and free boundary in one dimensional space. The main objective is to obtain the asymptotic behavior of spread of an invasive or new IG prey species via a free boundary. In two cases, we prove a spreading‐vanishing dichotomy for this model, specifically, the IG prey species either successfully spreads to infinity as t→∞ at the front and survives in the new environment or spreads within a bounded area and dies out in the long run. The long time behavior of (R,N,P) and criteria for spreading and vanishing are also obtained. And then, we estimate the asymptotic spreading speed of the free boundary when spreading happens. Besides, two numerical examples are given to illustrate the impacts of initial occupying area and expanding capability on the free boundary.     

Upscaling of solute transport in heterogeneous media with non-uniform flow and dispersion fields

An analytical and computational model for non-reactive solute transport in periodic heterogeneous media with arbitrary non-uniform flow and dispersion fields within the unit cell of length ε is described. The model lumps the effect of non-uniform flow and dispersion into an effective advection velocity V_e and an effective dispersion coefficient D_e. It is shown that both V_e and D_e are scale-dependent (dependent on the length scale of the microscopic heterogeneity, ε), dependent on the Péclet number P_e, and on a dimensionless parameter α that represents the effects of microscopic heterogeneity. The parameter α, confined to the range of [?0.5, 0.5] for the numerical example presented, depends on the flow direction and non-uniform flow and dispersion fields. Effective advection velocity V_e and dispersion coefficient D_e can be derived for any given flow and dispersion fields, and ε. Homogenized solutions describing the macroscopic variations can be obtained from the effective model. Solutions with sub-unit-cell accuracy can be constructed by homogenized solutions and its spatial derivatives. A numerical implementation of the model compared with direct numerical solutions using a fine grid, demonstrated that the new method was in good agreement with direct solutions, but with significant computational savings.     

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients.     

A mathematical model of imbibition phenomenon in heterogeneous porous media during secondary oil recovery process

The present paper discuses the solution of one-dimensional mathematical model for counter-current water imbibition phenomenon occurring into an oil-saturated cylindrical heterogeneous porous matrix. During secondary oil recovery process when water is injected in oil formatted heterogeneous porous matrix then at common interface the counter current imbibition phenomenon occurs due to the difference of viscosity of water and oil which satisfies imbibition condition _Vi=-_Vn$V_i=-V_n$. The governing differential equation of this phenomenon is in the form of non-linear partial differential equation which has been converted into non-linear ordinary differential by using similarity transformation. The solution of this problem has been obtained in term of power series by using appropriate boundary condition at common interface. The graphical presentation is obtained by using MATLAB and final solution physically interpreted.     

Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population

In this paper, we propose a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population, which is derived from the continuous case by using the well-known backward Euler method and by applying a Lyapunov function technique, which is a discrete version of that in the paper by Prüss et al. [J. Prüss, L. Pujo-Menjouet, G.F. Webb, R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B 6 (2006) 225-235]. It is shown that the global dynamics of this discrete epidemic model with latency are fully determined by a single threshold parameter.     

Traveling wave solutions in a diffusive system with two preys and one predator

This paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator. By constructing upper and lower solutions, the existence of nontrivial traveling wave solutions is established. The asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles. Applying the theory of asymptotic spreading, the nonexistence of traveling wave solutions is proved.     

Asymptotic electromagnetic fields in non-relativistic QED: The problem of existence revisited

This paper is devoted to the scattering of photons by charged particles in models of non-relativistic quantum mechanical matter coupled minimally to the soft modes of the quantized electromagnetic field. We prove existence of scattering states involving an arbitrary number of asymptotic photons of arbitrarily high energy. Previously, upper bounds on the photon energies seemed necessary in the case of n>1 asymptotic photons and non-confined, non-relativistic charged particles.     

Asymptotic behavior for the Stokes flow and Navier-Stokes equations in half spaces

Using the solution formula in Ukai (1987) [27] for the Stokes equations, we find asymptotic profiles of solutions in (n?2) for the Stokes flow and non-stationary Navier-Stokes equations. Since the projection operator is unbounded, we use a decomposition for P(u⋅∇u) to overcome the difficulty, and prove that the decay rate for the first derivatives of the strong solution u of the Navier-Stokes system in is controlled by for any t>0.     

Populations with individual variation in dispersal in heterogeneous environments: Dynamics and competition with simply diffusing populations

We consider a model for a population in a heterogeneous environment, with logistic-type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior has been observed in some natural systems. We study how environmental heterogeneity and the rates of switching and diffusion affect the persistence of the population. The reactiondiffusion systems in the models can be cooperative at some population densities and competitive at others. The results extend our previous work on similar models in homogeneous environments. We also consider competition between two populations that are ecologically identical, but where one population diffuses at a fixed rate and the other switches between two different diffusion rates. The motivation for that is to gain insight into when switching might be advantageous versus diffusing at a fixed rate. This is a variation on the classical results for ecologically identical competitors with differing fixed diffusion rates, where it is well known that "the slower diffuser wins".     

The Helmholtz equation in heterogeneous media: A priori bounds,well-posedness,and resonances

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation $\mathrm{?}?(A\mathrm{?}u)+k^{2}nu=?f$ where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are $L^{\infty }$ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be $C^0$ and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for $C^{\infty }$ convex interfaces with strictly positive curvature.     

Liouville-type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space:

Models of unidirectional propagation in heterogeneous excitable media

Two differential equation models of excitable media (threshold and recovery kinetics) with solutions that exhibit unidirectional propagation are presented. It is shown that unidirectional propagation in heterogeneous excitable media with non-oscillatory kinetics can be initiated from homogeneous initial data. Simulations on a reaction-diffusion model with FitzHugh-Nagumo kinetics and spatially heterogeneous parameters yields a rotating wave on a one-dimensional circular spatial domain. An ordinary differential equation model with four semi-coupled excitable cells and heterogeneous parameters is analyzed to determine a critical parameter region over which unidirectional propagation may occur.     

Traveling fronts in space–time periodic media

This paper is concerned with the existence of pulsating traveling fronts for the equation:

(1)

∂_tu−(A(t,x)u)+q(t,x)u=f(t,x,u),