The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

The role of the Crank-Gupta model in the theory of free and moving boundary problems

A very brief review is given of the striking way in which the Crank-Gupta model has enhanced our understanding of the well-posedness of free and moving boundary problems.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

Asymptotics for a free boundary model in price formation

We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria.The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.     

A free boundary problem for two-species competitive model in ecology

This paper deals with a free boundary problem which is used to describe the two-species competitive model in ecology. The existence and uniqueness of a global classical solution are given by invoking the Schauder fixed point theorem. We study the evolution of the free boundary problem and show that the free boundary problem is well posed.     

Two-magnon states in the one-dimensional isotropic heisenberg model with free boundary conditions

The eigenfunctions and eigenvalues of the energy of two-magnon states in the finite one-dimensional isotropic Heisenberg model S = 1/2 with free boundary conditions were found by solving the Schrödinger equation. The obtained solutions are single-parametric in contrast to two-parametric solutions in the model with cyclic boundary conditions. The amplitudes of the wave functions of coupled two-magnon states exponentially depend on both the distance between the flipped spins and the coordinate of the center of the complex. This leads to a localization of low energy complexes at the ends of the ferromagnetic chain.     

The indicatrix method in free boundary problems

Summary An approximate method for nonlinear problems with functional constraints is considered, in which the constraint in the whole domain is replaced by the constraint on a manifold of lower dimension. The stability criterion is introduced, and convergence theorems are proved for the onedimensional problem. Numerical results for the elastic-plastic torsion problem are given.     

A free boundary problem arising from DCIS mathematical model

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.     

The free streaming operator: A mathematical model for diffusive multiplying boundary conditions

We show that the free streaming operator with diffusive multiplying boundary conditions is the generator of a quasi-bounded semigroup. We also examine some spectral properties of such an operator.     

On the leading correction of the Thomas-Fermi model: Lower bound

Summary We prove that the quantum mechanical ground state energy of an atom with nuclear chargeZ can be bounded from below by the sum of the Thomas-Fermi energy of the problem plusq/8Z ² plus terms of ordero(Z ² ).     

Anharmonic states of the one-dimensional anisotropic Heisenberg model with free boundary conditions

States described by anharmonic functions are shown to exist in the one-dimensional anisotropic Heisenberg model with a finite number of spins 1/2 and free boundary conditions. Several such states are determined. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 489–497, December, 1998.     

A dengue fever model with free boundary incorporating the time-periodicity and spatial-heterogeneity

Propagation of dengue fever is characterized by periodicity and seasonality and further influenced by geographic heterogeneity. To account for these characteristics, we formulate a dengue model in a spatial-heterogeneous and time-periodic environment. Moreover, the free boundary is additionally incorporated into our model to reflect the boundary change of region where dengue virus spreads. Employing the properties of the contagion risk threshold, that is the spatial-temporal basic reproduction ratio, we derive some sufficient conditions regarding the vanishing and spreading of virus. Importantly, the long-time asymptotic behavior of solution is studied in depth when spreading happens. Our findings manifest that as time goes on, dengue virus will behave periodically when spreading. Finally, these phenomena are numerically simulated and epidemiologically explained.     

A free boundary arising from a model for nerve conduction

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions (Trans. Amer. Math. Soc.277 (1983), 1–42) introduced the class of “viscosity” solutions of these equations and proved uniqueness within this class. This paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness.     

A free boundary tumor model with time dependent nutritional supply

A non-autonomous free boundary model for tumor growth is studied. The model consists of a nonlinear reaction diffusion equation describing the distribution of vital nutrients in the tumor and a nonlinear integro-differential equation describing the evolution of the tumor size. First the global existence and uniqueness of a transient solution is established under some general conditions. Then with additional regularity assumptions on the consumption and proliferation rates, the existence and uniqueness of steady-state solutions is obtained. Furthermore the convergence of the transient solutions toward the steady-state solution is verified. Finally the long time behavior of the solutions is investigated by transforming the time-dependent domain to a fixed domain.     

The behavior of the free boundary near the fixed boundary for a minimization problem

We show that the free boundary ∂{u > 0}, arising from the minimizer(s) u, of the functional