Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem

We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character of the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.     

Incorporating inventory and routing costs in strategic location models

We consider a supply chain design problem where the decision maker needs to decide the number and locations of the distribution centers (DCs). Customers face random demand, and each DC maintains a certain amount of safety stock in order to achieve a certain service level for the customers it serves. The objective is to minimize the total cost that includes location costs and inventory costs at the DCs, and distribution costs in the supply chain. We show that this problem can be formulated as a nonlinear integer programming model, for which we propose a Lagrangian relaxation based solution algorithm. By exploring the structure of the problem, we find a low-order polynomial algorithm for the nonlinear integer programming problem that must be solved in solving the Lagrangian relaxation sub-problems. We present computational results for several instances of the problem with sizes ranging from 40 to 320 customers. Our results show the benefits of having an integrated supply chain design framework that includes location, inventory, and routing decisions in the same optimization model.     

Global Bifurcation on Time Scales

We consider the structure of the solution set of a nonlinear Sturm-Liouville boundary value problem defined on a general time scale. Using global bifurcation theory we show that unbounded continua of nontrivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and we show that certain nodal properties of the solutions are preserved along these continua. These results extend the well-known results of Rabinowitz for the case of Sturm-Liouville ordinary differential equations.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

Numerical analysis of a bilateral frictional contact problem for linearly elastic materials

We consider a mathematical model which describes the contactbetween a linearly elastic body and an obstacle, the so-calledfoundation. The process is quasistatic and the contact is bilateral,i.e. there is no loss of contact during the process. The frictionis modelled with Tresca's law. The variational formulation ofthe problem is a nonlinear evolutionary inequality for the displacementfield which has a unique solution under certain assumptionson the given data. We study spatially semi-discrete and fullydiscrete schemes for the problem with finite-difference discretizationin time and finite-element discretization in space. The numericalschemes have unique solutions. We show the convergence of thescheme under the basic solution regularity. Under appropriateregularity assumptions on the solution, we derive optimal ordererror estimates. Finally, we present numerical results in thestudy of two-dimensional test problems.     

Existence and quasilinearization in Banach spaces

We establish some existence results for the nonlinear problem Au=f in a reflexive Banach space V, without and with upper and lower solutions. We then consider the application of the quasilinearization method to the above mentioned problem. Under fairly general assumptions on the nonlinear operator A and the Banach space V, we show that this problem has a solution that can be obtained as the strong limit of two quadratically convergent monotone sequences of solutions of certain related linear equations.     

Exact and approximation algorithms for a soft rectangle packing problem

We consider the problem of finding an arrangement of rectangles with given areas that minimizes the total length of all inner and outer border lines. We present a polynomial time approximation algorithm and derive an upper bound estimation on its approximation ratio. Furthermore, we give a formulation of the problem as mixed-integer nonlinear program and show that it can be approximatively reformulated as linear mixed-integer program. On a test set of problem instances, we compare our approximation algorithm with another one from the literature. Using a standard numerical mixed-integer linear solver, we show that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality, or gives better primal and dual bounds if a certain time-limit is reached before.     

Collapse Rate of Solutions of the Cauchy Problem for the Nonlinear Schrödinger Equation

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.

     

Reduction from a Semi-Infinite Interval to a Finite Interval of a Nonlinear Boundary Value Problem for a System of Second-Order Equations with a Small Parameter

We consider a boundary value problem over a semi-infinite interval for a nonlinear autonomous system of second-order ordinary differential equations with a small parameter at the leading derivatives. We impose certain constraints on the Jacobian under which a solution to the problem exists and is unique. To transfer the boundary condition from infinity, we use the well-known approach that rests on distinguishing the variety of solutions satisfying the limit condition at infinity. To solve an auxiliary Cauchy problem, we apply expansions of a solution in the parameter.     

三阶两点边值问题解的存在性与唯一性

主要考虑如下三阶两点边值问题通过对非线性项f作适当的限制,利用上下解方法,获得三阶两点边值问题解的存在性结果.特别之处是,一个截断技巧和Nagumo条件的引入和使用.同时得到了解的唯一性结果.     

The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.     

On a system of nonlinear wave equations with Balakrishnan–Taylor damping

In this paper, we study the initial-boundary value problem for a coupled system of nonlinear viscoelastic wave equations of Kirchhoff type with Balakrishnan–Taylor damping terms. For certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. Also, we show that nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of stronger damping.     

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

A degenerate elliptic-parabolic system arising in competitive contaminant transport

The objective of this work is to study a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that this system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions, by addressing a more general problem. In addition, we conclude, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.     

Blow-up vs. spurious steady solutions

In this paper, we study the blow-up problem for positive solutions of a semidiscretization in space of the heat equation in one space dimension with a nonlinear flux boundary condition and a nonlinear absorption term in the equation. We obtain that, for a certain range of parameters, the continuous problem has blow-up solutions but the semidiscretization does not and the reason for this is that a spurious attractive steady solution appears.

     

奇摄动非线性系统初值问题的套层解

本文研究一类二阶非线性系统的初值问题的奇摄动,揭示了其解呈现双重初始层的性质,通过引进不同量级的伸长变量,得到解的一致有效的渐近展开式。     

On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium

We consider a system of nonlinear coupled partial differential equations that models immiscible two-phase flow through a porous medium. A primary difficulty with this problem is its degenerate nature. Under reasonable assumptions on the data, and for appropriate boundary and initial conditions, we prove the existence of a weak solution to the problem, in a certain sense, using a compactness argument. This is accomplished by regularizing the problem and proving that the regularized problem has a unique solution which is bounded independently of the regularization parameter. We also establish a priori estimates for uniqueness of a solution.     

通过移动媒质避免爆破

This paper deals with two parabolic initial-boundary value problems in multidimensional domain. The first problem describes the situation where the spherical medium is static and the nonlinear reaction takes place only at a single point. We show that under some conditions, the solution blows up in finite time and the blow-up set is the whole spherical medium. When the spherical medium is allowed to move in a special space, we investigate another parabolic initial-boundary value problem. It is proved that the blow-up can be avoided if the acceleration of the motion satisfies certain conditions.