Mathematical Models and Methods in Applied Sciences

An optimal harvesting problem for a parabolic partial differential system modeling two subpopulations of the same species is investigated. The two subpopulations are competing for resources. Under conditions on the smallness of the time interval and certain biological parameters, existence and uniqueness of an optimal control pair are established.

This paper is to study the asymptotic behavior of solutions for an initial–boundary value problem to Jin–Xin's 2×2 relaxation hyperbolic system. When the initial data are small perturbation of the superposition of two travelling waves at t=0, subsequent to the previous work,⁶ we further show the convergence rates of the IBVP solutions to the superposition of two waves. Precisely, when the initial perturbations decay in the exponential or algebraic forms, we prove that the corresponding solutions tend to the superposition of two waves time-asymptotically in the exponential or algebraic forms, respectively. The method adopted is the weighted energy method. The use of a shift function for the forward travelling wave and the special choice of shift functions for backward travelling plays a key role to overcome the difficulties caused by the boundary and degeneration.

We consider a Galerkin finite element method that uses bilinear elements on a class of Shishkin-type meshes for a model singularly perturbed convection–diffusion problem. The recovered gradient is itself piecewise bilinear, with values at the nodes obtained by first interpolating the gradient of the finite element approximation at the centroids of the elements sharing the node. We prove a superconvergence estimate for the recovered gradient uniformly with respect to the singular perturbation parameter. Numerical experiments support our results.

The goal of this paper is to give a systematic analysis of compactness properties for transport equations with general boundary conditions where an abstract boundary operator relates the incoming and outgoing fluxes. The analysis involves two parameters: The velocity measure and the collision operator. Hence, for a large class of (velocity) measures and under appropriate assumptions on scattering operators compactness results are obtained. Using the positivity (in the lattice sense) and the comparison arguments by Dodds–Fremlin, their converses are derived, and necessary conditions for some remainder term of the Dyson–Phillips expansion to be compact are given. Our results are independent of the properties of the boundary operators and play a crucial role in the understanding of the time asymptotic structure of evolution transport problems. Also, although solutions of transport equations propagate singularities (due to the hyperbolic nature of the operator), they bring the regularity in the variable space (regardless of the boundary operator). We end the paper by applying the obtained results to discuss the existence of solutions to a nonlinear boundary value problem and to describe in detail the various essential spectra of transport operators with abstract boundary conditions.

In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists of describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced system with a few "global" variables.

We present a time varying discrete model in which we distinguish two processes with different time scales. By defining the global variables as appropriate linear combinations of the state variables, we transform the system into a reduced one. The variables corresponding to the original and reduced systems can be related, therefore allowing one the study of the former in terms of the latter. The property of weak ergodicity, which has to do with the capacity of a system to become asymptotically independent of initial conditions, is explored for the original and reduced systems.

The general method is also applied to aggregate a time-dependent multiregional model which appears in the field of population dynamics in two different cases: Fast migration with respect to demography and fast demography with respect to migration.

A one-dimensional model of a curved rod is derived from the three-dimensional linearized elasticity. No positivity assumption on the curvature of the central line of the curved rod is made. The model is obtained by taking the limit in the equilibrium equation of the three-dimensional elastic rod when the thickness of the rod goes to zero. The appropriate convergence result is proved.

The high field limit for the semiconductor Boltzmann equation with Pauli exclusion terms is investigated. The limit problem is shown to have a unique solution for every given density. The proof relies on a linearization procedure together with a continuation argument. The density is finally proven to converge in the high field limit towards the solution of a nonlinear hyperbolic equation.

A model for a continuum gas flowing through a porous matrix is proposed where the gas kinetics is governed by the Boltzmann equation and the solid phase by the energy equation. In the Boltzmann equation the integral relative to the gas–solid collisions is evaluated as for the collisions of hard spheres molecules against much heavier and longer straight particles (Lebowitz model of a sticks gas), randomly distributed in space according to a Maxwellian function with zero mean velocity. The mean flow is one-dimensional but the molecules are free to move in all three space dimensions. In the continuum limit, the moments of the Boltzmann equation provide the mass continuity, energy and momentum equations, the last one expressing the Darcy law for a compressible gas. The transport coefficients are analytically evaluated and a few examples are dealt with.

We consider a linear dissipative wave equation in ℝ^N with periodic coefficients. By means of Bloch wave decomposition, we obtain an expansion of solutions as t→∞ and conclude that, in a first approximation, the solutions behave as the homogenized heat kernel.