Mathematical Models and Methods in Applied Sciences

The paper is devoted to the analysis of a drift-diffusion-Schrödinger–Poisson (DDSP) system. From the physical point of view, it describes the transport of a quasi-bidimensional electron gas confined in a nanostructure. Existence, uniqueness and long-time behavior of a weak solution were already obtained in Ref. 8 for constant scalar diffusion matrices. Here, we develop an L log L existence theory for the DDSP system for a general class of smooth diffusion matrices. Our argument relies on a Trudinger estimate for the entropy functional and a sharp bound on the Hamiltonian's spectrum.

In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L²-energy method, L¹-estimate and the monotonicity of profile obtained by the maximum principle.

For the scalar wave equation in periodic structures, we establish rigorously the manifestation of a spectral bandgap as a frequency interval of exponentially small transmission of energy through a slab of the crystal when the thickness of the slab tends to infinity. We extend the result to slabs that contain a planar defect for frequencies that do not coincide with guided mode frequencies for the defect. For one-dimensional crystals, we prove that the transmission approaches a nonzero value at guided mode frequencies and give an explicit formula for the transmission. The main analytical tool in the multidimensional case is the Calderón boundary integral projectors for the Helmholtz equation.

A Boltzmann transport model for dose calculation in radiation therapy is considered. We formulate an optimal control problem for the desired dose. We prove existence and uniqueness of a minimizer. Based on this model, we derive optimality conditions. The P_N discretization in angle of the full model is considered. We show that the P_N approximation of the optimality system is in fact the optimality system of the P_N approximation, provided that, instead of the usually used Marshak boundary conditions, Mark's boundary conditions are used. Numerical results in one and two dimensions are presented.

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.