On the dynamics of radially symmetric granulomas

A granuloma is a collection of macrophages that contains bacteria or other foreign substances that the body?s immune response is unable to eliminate. In this paper we present a simple mathematical model of radially symmetric granuloma dynamics. The model consists of a coupled system of two semi-linear parabolic equations for the macrophage density, and the bacterial density. The boundary of the granuloma is free. This simple framework makes it possible to conduct a mathematical analysis of the system dynamics. In particular, we show that the model system has a unique solution, and that, depending on the biological parameters; the bacterial load either disappears over time or persists. We use numerical methods to establish the existence of stationary solutions and examine how a stationary solution changes with the reproductive rate of the bacteria. These simulations show that the structure of the granuloma breaks down as the reproductive rate of the bacteria increases.     

Debt–equity swap with finite time horizon—variational inequality approach

This paper concerns the finite-horizon optimal reorganization problem under debt–equity swap. The model of equity is formulated as a parabolic variational inequality, or equivalently, a free boundary problem, where the free boundary corresponds to the optimal reorganization boundary. The existence and uniqueness of the solution are proven and the behavior of the free boundary, such as smoothness, monotonicity and boundedness, is studied. To the best of our knowledge, this is the first complete set of results on debt–equity swap for finite maturity obtained using PDE techniques.     

Relaxation-time limit and initial layer in the isentropic hydrodynamic model for semiconductors

This work is concerned with the relaxation-time limit in the multidimensional isentropic hydrodynamic model for semiconductors in the critical Besov space. Firstly, we construct formal approximations of the initial layer solution to the nonlinear problem by the matched expansion method. Then, assuming some regularity of the solution to the reduced problem, and proves the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval.     

The Landau–Hall problem on canal surfaces

In this note we study the Landau–Hall problem on a generalized canal surface and classify all uniform magnetic trajectories of a charged particle moving on such a surface under the action of a uniform magnetic field.     

Improved trial methods for a class of generalized Bernoulli problems

The aim of this article is to develop improved trial methods for the solution of a generalized exterior Bernoulli free boundary problem. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. Appropriate update rules are obtained by expanding the state's Dirichlet data at the actual boundary via a Taylor expansion of first and second order. The resulting trial methods converge linearly for both cases, although the trial method based on the second order Taylor expansion is much more robust. Nevertheless, via results of shape sensitivity analysis, we are able to modify the update rules such that their convergence is improved. The feasibility of the proposed trial methods and their performance is demonstrated by numerical results.     

A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p  -Laplacian, p>1$p>1$, on bounded cylindrical domains of R^N+1${\mathbb{R}}^{N+1}$,

_∂tu−_Δpu−λ|u|^p−2u≥0,${\partial }_{t}u-{\mathrm{\Delta }}_{p}u-\lambda {|u|}^{p-2}u\ge 0,$

with nonnegative initial–boundary conditions and λ≤0$\lambda \le 0$, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<2$1<p<2$, and give a counterexample to it for p>2$p>2$. Also the Weak Maximum Principle for λ≤_λ1$\lambda \le {\lambda }_1$ is established.