Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives

This paper concerns a class of nonlinear variational problems involving pointwise constraints on the second derivatives. Our aim is to describe the set of data for which these problems have solutions and to analyse the structure of the set of solutions under suitable assumptions on the asymptotic behaviour of the nonlinear term. In particular, if this term is assumed to be convex, then we can specify the number of solutions and obtain exact multiplicity results.The existence, nonexistence and multiplicity results we obtain show that the presence of constraints of this kind produces some phenomena which are typical of nonlinear elliptic equations with “jumping” nonlinearities.     

A global attractor for the plate equation with displacement-dependent damping

We study the long-time behaviour of solutions of the plate equation with nonlinear damping coefficient. We prove under suitable conditions that this equation possesses a global attractor.     

Positive solutions for a Dirichlet problem

1. IntroductionSince the work of Ambrosetti and Rabinowitz[l], the problems similar to{;t2:<::l">, (l.l)have been studied extensively But it is well known that, for applying the Moulltain PassTheOrem, we atway8 assum that g(x, 8) is suPerlineax in s at indnity; moeove) a strongercondition like (AR) (see later on) is required. If these conditions are not satisfied, can wealso get solutions for problem (1.1) by a Mountain Pass Theorem? So, ill this paPer, westudy the following Dirichlet pr…     

具有Hardy奇异项的共振半线性椭圆方程的解

利用变分法研究了具有Dirichlet边值问题-△u-μ(u/(|x|²))=f(x,u)的解的存在性问题,在适当的条件下给出了其解的存在性定理.     

Gauss–Seidel method for multi-valued inclusions with Z mappings

We consider a problem of solution of a multi-valued inclusion on a cone segment. In the case where the underlying mapping possesses Z type properties we suggest an extension of Gauss–Seidel algorithms from nonlinear equations. We prove convergence of a modified double iteration process under rather mild additional assumptions. Some results of numerical experiments are also presented.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

高阶波动方程的时空估计与低能量散射

本文研究了高阶波动方程的低能量散射理论，基本工具是高阶线性波动方程解的时空估计．与经典的二阶波动方程解的时空估计证明不同，我们采用泛函分析的方法与待定指标技巧，首次给出了高阶线性波动方程的时空估计，藉此与非线性函数在齐次Ｓｏｂｏｌｅｖ空间中的估计，获得了高阶波动方程的低能量散射结论．与此同时，也得到了具临界增长的高阶波动方程的柯西问题在低能量条件下的整体存在唯一性．     

Analysis of a nonlinear model of the vibrating string

A nonlinear model of the vibrating string is studied under the assumption that the motion is transversal and existence and uniqueness theorems are given for the Cauchy-Dirichlet problems. Some numerical experiments are also described, illustrating the behaviour of this model with respect to the nonlinear Kirchhoff model and the classical linear model of D'Alembert.The research has been supported by MURST 40% and 60% Research Contracts     

奇异三阶两点边值问题解的一个存在定理

By constructing suitable Banach space.an existence theorem is established under a condition of linear growth for the third-order boundary value problem u'"(t) f(t,u(t),t'(t),u"(t))=0,0＜t＜1,u(0)=u'(0)=u'(1)=0,where the nonlinear term contains first and second derivatives of unknown function.In this theorem the nonlinear term f(t,u,v,w)may be singular at t=0 and t=1.The main ingredient is Leray-Schauder nonlinear alternative.     

MULTIPLE SOLUTIONS TO A SECOND-ORDER m-POINT BOUNDARY VALUE PROBLEM

In this paper, we are concerned a class of second-order m-point boundary value problem. The existence results of at least three positive solutions are given by using a fixed-point theorem and imposing growth conditions on the nonlinear term, which depends on the first derivative.     

Properties of a family of merit functions and a merit function method for the NCP

A family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given.     

Positive solutions of nonlinear beam equations with time and space singularities

We consider the existence and multiplicity of positive solutions to a nonlinear fourth-order two-point boundary value problem. The nonlinear term may be singular with respect to both the time and space variables. In mechanics, the problem describes the deformation of an elastic beam fixed at the left and supported at the right by sliding clamps. By introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several local existence theorems are obtained.     

On Numerov's method for a class of strongly nonlinear two-point boundary value problems

The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.     

关于一类具阻尼项的非线性波方程的Cauchy问题

The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the above-mentioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.     

A NONLOCAL NONLINEAR BOUNDARYVALUE PROBLEM FOR THE HEAT EQUATIONS

51.IntroductionandMalnResultsLetflCR"(n=2or3)beaboundedsmoothdomainwithboundaryan=r1ur2,wherer1nr2=Oandr1,r2/O.Weconsiderthefollowingmixedinitial-boundaryvalueproblemfortheheatequation'whereH'DHCR-Risastrict1ydecreasingtunction,ustancnormalvectorattheboundary.Theboundaryconditiononr2appearsinthermistors.Howison,RodriguesandShillorl1]studiedsuchkindofboundaryvalueproblemforellipticequations.HerewesupposethatHisonto'nH=R,andwherea,bdenote,andwillalwaysdenote,varieskindofconstants.Underhy…     

On the finite time blow-up for filtration problems with nonlinear reaction

We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow-up of solutions subject to sufficiently large initial data provided that the reaction term “overpowers” the nonlinear diffusion in a certain sense. Secondly, under related assumptions on the nonlinearities, we show that initial data above positive stationary state solutions will always lead to finite time blow-up.     

一个求解P0函数非线性互补问题的非内部连续化算法

基于黄正海等2001年提出的光滑函数,本文给出一个求解P0函数非线性互补问题的非内部连续化算法.所给算法拥有一些好的特性.在较弱的条件下,证明了所给算法或者是全局线性收敛,或者是全局和局部超线性收敛.给出了所给算法求解两个标准测试问题的数值试验结果.     

非线性一维p-Laplace方程的两正解存在定理

考察了一类非线性一维p-Laplace方程正解的多解性.主要结论表明,即使非线性项在0点和无穷远处不满足通常的增长条件,该方程仍可能有两个正解.