Mathematical Models and Methods in Applied Sciences

This work is devoted to the study of a singular reaction–diffusion system arising in modelling the introduction of a lethal pathogen within an invading host population. In the absence of the pathogen, the host population exhibits a bistable dynamics (or Allee effect). Earlier numerical simulations of the singular SI model under consideration have exhibited stable travelling waves and also, under some circumstances, a reversal of the wave front speed due to the introduction of the pathogen. Here we prove the existence of such travelling wave solutions, study their linear stability and give analytical conditions yielding a reversal of the wave front speed, i.e. the invading host population may eventually retreat following the introduction of the lethal pathogen.

A Walrasian pure exchange economy with utility functions, a particular case of a general economic equilibrium problem, is considered in this paper. We assume that each agent is endowed with goods and maximizes his utility function, under his budget constraints. We are able to characterize the Walrasian equilibria as solution of an associated quasi-variational inequality. This approach allows us to obtain an existence result of equilibrium solutions. As an application, we provide the explicit equilibrium in the case of two agents and two goods.

We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 - d inviscid Burgers equation. We first prove the existence of minimizers and, by a Γ-convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so-called one-sided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches: the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a solution we propose a new method, that we shall call alternating descent method, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. This method distinguishes and alternates the descent directions that move the shock and those that perturb the profile of the solution away of it producing very efficient and fast descent algorithms.

We consider a one-dimensional model of generally curved elastic arches whose cross-sections are rectangular. The model is of Naghdi's type which is a generalization of the Timoshenko beam model, which allows bending, membrane and transverse shearing deformations. Its form is basically determined in the literature, except for the value of a shear correction factor. With this factor being set to 1, we prove that the modelling error in the interior relative energy norm is proportional to the arch thickness. This result holds for the full range of arch shapes and very general loads. Lower modelling accuracy is proven to hold up to the arch ends. Any shear correction factor other than 1 makes the model diverge from the elasticity theory when a significant shear is involved in the deformation.

In this paper a time-dependent as well as a stationary drift-diffusion-Poisson system for semiconductors are studied. Global existence and uniqueness of weak solution of the time-dependent problem are proven and we also prove the existence and uniqueness of the steady state. It is shown that as time tends to infinity, the solution of the time-dependent problem will converge to a unique equilibrium. Due to the presence of recombination-generation rate R in our drift-diffusion-Poisson model, the work of this paper in some sense extends the results in the previous literature (on both time-dependent problem and stationary problem).