A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R⁺. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L¹ norm was used, we apply the flat metric, similar to the Wasserstein W₁ distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.     

Mathematical analysis of radionuclides displacement in porous media with nonlinear adsorption

This paper is devoted to the mathematical analysis of the miscible displacement of a set of radionuclides in a flow occurring in a heterogeneous porous medium. The flow is governed by Darcy's law, and the motion of the chemical species is given by a nonclassical advection-diffusion-reaction equations system. The novelty of the model lies in the adsorption phenomenon that leads to a time derivative of a nonlinear term in these equations. A semi-discretization method is used to establish the existence of weak solutions to this system. Uniform L^∞-estimates on the solutions are specified.     

Existence and stability of travelling waves in fixed-bed reactors

The model equations of the catalytic fixed-bed reactor often possess solutions in the form of travelling wave fronts similar to the well-known case of Fisher's equation. The mathematical investigation of these waves requires searching for solutions of singular boundary value problems in the phase plane or in the three-dimensional phase space. In this paper necesary and sufficient conditions are derived which are to be satisfied by the model parameters and the propagation velocity of the wave front if wave solutions exist. Moreover, sufficient conditions for the asymptotic stability of these solutions are proved where the perturbations are supposed to belong to a certain weighted L²-space. Finally, the connection between the initial distribution of the state variable and the velocity of the wave is discussed.     

A mathematical and physical Study of multiscale deconvolution models of turbulence

This work presents a rigorous analysis of mathematical and physical properties for solutions of multiscale deconvolution turbulence models. We show that solutions of these models exactly conserve model quantities for the integral invariants of fundamental physical importance: kinetic energy, helicity, and (in two dimensions) enstrophy. The kinetic energy conservation is the key that allows us to next apply the phenomenology of homogeneous, isotropic turbulence to establish the existence of a model energy cascade and, in particular, that the cascade exhibits enhanced energy dissipation in a secondary accelerated cascade, which ends at the model's microscale (which we establish is larger than the Kolmogorov microscale). We also prove that the model dissipates energy at the same rate as true turbulent flow, ～ O(U³ ∕ L), independent of Reynolds number. Lastly, we prove the existence of global attractors for the model solutions; the proof of which also shows that solutions are actually one degree of regularity higher than previously known. Copyright © 2012 John Wiley & Sons, Ltd.     

A priori estimates of higher order derivatives of solutions to the FitzHugh-Nagumo equations

In this paper we establish L^∞-bounds for the derivatives of all orders of the solutions to the FitzHugh-Nagumo equations, by means of comparison functions. We obtain bounds for the initial value problem, the Dirichlet problem and the Neumann problem. The FitzHugh-Nagumo equations arise in mathematical biology as a model for the conduction of electrical impulses along a nerve axon.     

On the Well-posedness and Asymptotic Behavior of a Nonlinear Dispersive System in Weighted Spaces

This paper is concerned with a model for propagation of long waves in a channel generated by a wave maker mounted at one end. The mathematical structure consists in a coupled system of two nonlinear Korteweg-de Vries equations posed on the positive half line. Under the effect of a localized damping term it is shown that the solutions of the system are exponentially stable and globally well-posed in the weighted space L ²((x+1) ^m dx) for m≥1. The stabilization problem is studied constructing a Lyapunov function by induction on m and the well-posedness is obtained by passing to the limit in a sequence of solutions in L ²(e ^2bx dx) for b>0.     

Global solutions for a general strongly coupled prey–predator model

This work investigates global solutions for a general strongly coupled prey–predator model that involves (self-)diffusion and cross-diffusion, where the cross-diffusion is of the form v/(1+u^ℓ) with ℓ≥1. Very few mathematical results are known for such models, especially in higher spatial dimensions.     

A model combining acid-mediated tumour invasion and nutrient dynamics

We illustrate a mathematical model for the evolution of multicellular tumour spheroids in a host tissue, including the effect of the excess H⁺ ions and the nutrient dynamics. Both the avascular and the vascular case are considered. The model is a nontrivial generalization of the simple scheme proposed in [K. Smallbone, D.J. Gavaghan, R.A. Gatenby, P.K. Maini, The role of acidity in solid tumour growth and invasion, J. Theoret. Biol. 235 (2005) 476–484]. Many different situations may occur, depending on the values of the physical parameters involved. Existence of solutions and qualitative properties are investigated.     

On explicit L∞(QT)‐estimates for a model of tissue invasion by solid tumors

In this paper, we prove that if the initial data is small enough, we obtain an explicit L^∞(Q_T)‐estimate for a two‐dimensional mathematical model of cancer invasion, proving an explicit bound with respect to time T for the estimate of solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Positive computational modelling of the dynamics of active and inert biomass with extracellular polymeric substances

Departing from a complex system of nonlinear partial differential equations that models the growth dynamics of biological films, we provide a finite-difference model to approximate its solutions. The variables of interest are measured in absolute scales, whence the need of preserving the positivity of the solutions is a mathematical constraint that must be observed. In this work, we provide a numerical discretization of our mathematical model which is capable of preserving the non-negative character of approximations under suitable conditions on the model and computational parameters. As opposed to the nonlinear model which motivates this report, our numerical technique is a linear method which, under suitable circumstances, may be represented by an M-matrix. The fact that our method is a positivity-preserving scheme is established using the inverse-positive properties of these matrices. Computer simulations corroborate the validity of the theoretical findings.     

Existence and uniqueness of weak solutions for precipitation fronts: A novel hyperbolic free boundary problem in several space variables

The determination of the large‐scale boundaries between moist and dry regions is an important problem in contemporary meteorology. These phenomena have been addressed recently in a simplified tropical climate model through a novel hyperbolic free boundary formulation yielding three families (drying, slow moistening, and fast moistening) of precipitation fronts. The last two wave types violate Lax's shock inequalities yet are robustly realized. This formal hyperbolic free boundary problem is given here a rigorous mathematical basis by establishing the existence and uniqueness of suitable weak solutions arising in the zero relaxation limit. A new L²‐contraction estimate is also established at positive relaxation values. © 2010 Wiley Periodicals, Inc.     

Time reversal for modified oscillators

We consider a new completely integrable case of the time-dependent Schrödinger equation in ®ⁿ with variable coefficients for a modified oscillator that is dual (with respect to time reversal) to a model of the quantum oscillator. We find a second pair of dual Hamiltonians in the momentum representation. The examples considered show that in mathematical physics and quantum mechanics, a change in the time direction may require a total change of the system dynamics to return the system to its original quantum state. We obtain particular solutions of the corresponding nonlinear Schrödinger equations. We also consider a Hamiltonian structure of the classical integrable problem and its quantization.     

Identification of Cl(Ca) channel distributions in olfactory cilia

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. Transduction of an odor into an electrical signal occurs in the membranes of the cilia. The Cl(Ca) channels that reside in the ciliary membrane are activated by calcium, allow a depolarizing efflux of Cl⁻ and are thought to amplify the electrical signal to the brain. In this paper, a mathematical model consisting of partial differential equations is developed to study two different experiments; one involving the interaction of the cyclic-nucleotide-gated (CNG) and Cl(Ca) channels and the other, the diffusion of Ca²⁺ into cilia. This work builds on an earlier study (Mathematical modeling of the Cl(Ca) ion channels in frog olfactory cilia. Ph.D. Thesis, University of Cincinnati, Cincinnati, OH, 2006; Math. Comput. Modelling 2006; 43 :945–956; Biophys. J. 2006; 91 :179–188), which suggested that the CNG channels are clustered at about 0.28 of the length of a cilium from its open end. Closed-form solutions are derived after certain reductions in the model are made. These special solutions provide estimates of the channel distributions. Scientific computation is also used. This preliminary study suggests that the Cl(Ca) ion channels are also clustered at about one-third of the length of a cilium from its open end. Copyright © 2008 John Wiley & Sons, Ltd.     

Distributed model of a self-developing market economy

Fairly rigorous mathematical laws are applied to construct a mathematical model of a self-developing market economy with movement of time-dependent capital in the technology space. The model is a system of nonlinear partial differential equations. We analyze the bifurcations of the spatially homogeneous solutions that describe the dynamics of macro-variables of the economic system. We also study some properties of the solutions of the partial differential equations that follow from diffusion of capital and consumer demand.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 243–262, 2002.     

MIP approaches for the integrated berth allocation and quay crane assignment and scheduling problem

In this paper we consider an integrated berth allocation and quay crane assignment and scheduling problem motivated by a real case where a heterogeneous set of cranes is considered. A first mathematical model based on the relative position formulation (RPF) for the berth allocation aspects is presented. Then, a new model is introduced to avoid the big-M constraints included in the RPF. This model results from a discretization of the time and space variables. For the new discretized model several enhancements, such as valid inequalities, are introduced. In order to derive good feasible solutions, a rolling horizon heuristic (RHH) is presented. A branch and cut approach that uses the enhanced discretized model and incorporates the upper bounds provided by the RHH solution is proposed. Computational tests are reported to show (i) the quality of the linear relaxation of the enhanced models; (ii) the effectiveness of the exact approach to solve to optimality a set of real instances; and (iii) the scalability of the RHH based on the enhanced mathematical model which is able to provide good feasible solutions for large size instances.     

集值均衡问题近似Benson真有效解的非线性刻画

在一般的数学模型中,由于要忽略一些次要因素,所建的模型往往是近似的,且对数学模型利用数值算法所求得的解大多是近似解。另一方面,在可行集非紧的情况下,精确解的解集往往是空集,而在较弱的条件下近似解集可以是非空的。在Hausdorff局部凸拓扑线性空间中分别研究了无约束和带约束集值均衡问题近似Benson真有效解。在没有任何凸性假设下,利用非线性泛函分别建立了最优性条件。     

Post-surgical passive response of local environment to primary tumor removal

Prompted by recent clinical observations on the phenomenon of metastasis inhibition by an angiogenesis inhibitor, a mathematical model is developed to describe the post-surgical response of the local environment to the “surgical” removal of a spherical tumor in an infinite homogeneous domain. The primary tumor is postulated to be a source of growth inhibitor prior to its removal at t = 0; the resulting relaxation wave arriving from the disturbed (previously steady) state is studied, closed form analytic solutions are derived, and the asymptotic speed of the pulse is estimated to be about 2 × 10⁻⁴ cm/sec for the parameters chosen. In general, the asymptotic speed is found to be 2√Dγ, where D is the diffusion coefficient and γ is the inhibitor depletion or decay rate.     

Positive solutions of an elliptic partial differential equation on R

In this paper, we discuss the existence, nonexistence and uniqueness of positive solutions of a one-parameter family of elliptic partial differential equations on R^N (N>2). These equations are of interests in mathematical biology and Riemannian geometry. Our approach are based on variational arguments and comparison principles.     

On the structural stability of thermoelastic model of porous media

In the present paper we study the structural stability of the mathematical model of the linear thermoelastic materials with voids. We prove that the solutions of problems depend continuously on the constitutive quantities, which may be subjected to error or perturbations in the mathematical modelling process. Thus, we assume to have changes in the various coupling coefficients of the model and then we establish estimates of continuous dependence of solutions. We have to outline that such estimates play a central role in obtaining approximations to these kinds of problems. To derive a priori estimates for a solution we first establish appropriate bounds for the solutions of certain auxiliary problems. These are achieved by means of so‐called Rellich‐like identities. We also investigate how the solution in the coupled model behaves as some coupling coefficients tend to zero. Copyright © 2007 John Wiley & Sons, Ltd.