A note on gradient blowup rate of the inhomogeneous hamilton-jacobi equations

The gradient blowup of the equation u_t = Δu + a(x)|∇u|^p + h(x), where p > 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate for radial solutions is established under the assumptions on the initial data so that the solution is monotonically increasing in time.     

Hamilton-Jacobi方程的小波Galerkin方法

本文选择Daubechies小波尺度函数空间作为Galerkin方法的测试函数空间,并将其应用于Hamilton-Jacobi方程,得到了求解Hamilton-Jacobi方程的小波Galerkin方法的数值格式．由于小波在时间和频率上的局部性,本方法适用于处理具有奇异解的问题,可以有效地防止数值振荡．数值试验显示,本方法是有效的．     

Hamilton-Jacobi方程的单调有限元格式

A monotone finite element scheme is obtained by applying the finite element method to the viscosity equation of the Hamilton-Jacobi equation on unstructured meshes. Under some constraints, we show that this scheme is monotone and its numerical solution converges to the viscosity solution of the Hamilton-Jacobi equa-tion. Numerical examples test the stability and the convergence of this scheme.     

求解Hamilton-Jacobi方程的有限元方法

本文将Galerkin二次有限元应于Hamilton-Jacobi方程，得到了求解Hamilton-Jacobi方程的数值格式。这些格式是TVD型的，在更强的条件下，基半离散格式的数值解收敛于Hamilton-Jacobi方程的粘性解。数值结果表明这类格式具有较高分辨导数间断的能力。     

用基于水平集方法的自适应运动网格变形方法解Hamilton-Jacobi方程组

A numerical scheme is presented for solving the Hamilton-Jacobi equation by applying adaptive moving grid methods of level-set-based deformation methods. Two numerical examples are given, which demonstrate the accuracy and efficiency of computing“extreme”and“spikes”of solutions to the Hamilton-Jacobi equation.     

THE LARGE TIME GENERIC FORM OF THE SOLUTION TO HAMILTON-JACOBI EQUATIONS

We use Hopf-Lax formula to study local regularity of solution to Hamilton-Jacobi (HJ) equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to We use Hopf-Lax formula to study local regularity of solution to Hamilton-Jacobi (HJ) equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T > 0, which depends only on the Hamiltonian and initial datum, for t > T the solution of the IVP (1.1) is smooth except for a smooth n-dimensional hypersurface, across which Du(x, t) is discontinuous. And we show that the hypersurface tends asymptotically to a given hypersurface with rate t 1 4 .HJ equation, i.e. for most initial data there exists a constant T > 0, which depends only on the Hamiltonian and initial datum, for t > T the solution of the IVP (1.1) is smooth except for a smooth n-dimensional hypersurface, across which Du(x, t) is discontinuous. And we show that the hypersurface tends asymptotically to a given hypersurface with rate t-1/4 .     

A NOTE ON THE GRADIENT PROJECTION METHOD WITH EXACT STEPSIZE RULE

In this paper, we give some convergence results on the gradient projection method with exact stepsize rule for solving the minimization problem with convex constraints. Especially, we show that if the objective function is convex and its gradient is Lipschitz continuous, then the whole sequence of iterations produced by this method with bounded exact stepsizes converges to a solution of the concerned problem.     

BLOWUP OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEARWAVE EQUATIONS

gi. IntroductionThis paper deals with solutionS of certain nonlinear wave equationS Of the formcorresponding to Antial conditionSwuersis the wave OPerstor.we are interested in showing the ~ up" Of solutions to (1.1)--(1.2). For that, wereIf ac ~ 1)(n ~ 1) > 2, global solutions of ~ equation subject to very general perturbationsof order p exist Provided the initial data are swhciently small (see I6] and references therein).We are also interested in esthaattw the take when "blow up" occurs. …     

ON THE RATE OF CONVERGENCE OF SOLUTIONS FOR A CLASS OF DIFFERENCE EQUATIONS

1IntroductionConsiderthefollowingdifferenceequationwhereGfRxR-Riscontinuous,p:Z -R ,k;Z -Z ,p(n)5M,k(n)5k,MER ,kEZ ;ZandZ denotethesetsofintegersandnonnegativeintegers,respective1y.Forintegersaand5,a>6,wedefineZ(a,5)={a,a l,'',6},Z(a)={a,a l,'.}.Weassumethat(i)G(x,.)ismonotonenon-decreasing,(ii)G(.,y)ismonotonenon-increasing;(iii)G(x,x)=o,G(x,y)5oforx>yandG(x,y)2oforx相似文献   

BLOWUP OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEARWAVE EQUATIONS

The author proves blow up of solutions to the Cauchy problem of certain nonlinear wave equations and, also, estimates the time when the blow up occurs.     

THE BLOWUP OF RADIALLY SYMMETRIC SOLUTIONS FOR 2-D QUASILINEAR WAVE EQUATIONS WITH CUBIC NONLINEARITY

61.IntroductionInthispaper,weconsiderthefollowingtwodimensionalquasilinearwaveequationswiththenonlinearityofcubicform:wherex=(x1,x2),E>Oissmallenough,c'(otu,7u)=c'(otu,Oru)=l a,(otu)' a2Ofuoru a,(oru)' o(Iotul' lorul'),f(otu'Vu)=f(otu,o'u)=b,(otu)' b,(o,u)'oru b,otu(oru)' b,(oru)' O(Iotul' loruI'),a1-a2 a3/o,uo(x),ul(x)areCooradialfunctions(thatis,smoothfunctionsoflx1')andsupportedinaffeedba.llofradiusM.Moreoveruo(x)/Ooru1(x)*O.OuraimistostudythelifespanTeofsolutionsto(l.1)andthebreakdow…     

GLOBAL BLOWUP AND BLOWUP RATE ESTIMATES OF SOLUTIONS FOR A CLASS OF NONLINEAR NON-LOCAL REACTION-DIFFUSION PROBLEMS

In this paper, the global blowup properties of solutions for a class of non-linear non-local reaction-diffusion problems are investigated by the methods of the priorestimates. Moreover, the blowup rate estimate of the solution is given.     

HERMITE WENO SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS FOR HAMILTON-JACOBI EQUATIONS

In this paper, we use Hermite weighted essentially non-oscillatory （HWENO） schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes （HWENO-RK） of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.     

A NATURAL GRADIENT ALGORITHM FOR THE SOLUTION OF LYAPUNOV EQUATIONS BASED ON THE GEODESIC DISTANCE

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.     

ON THE UPPER ESTIMATES OF FUNDAMENTAL SOLUTIONS OF PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

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非齐次对称特征值问题

引言 用SR~(n×n)表示所有。n×n实对称矩阵的集合。R~n表示n维线性空间。||·||_2表示向量的Euclid范数或矩阵的谱范数。 本文研究如下问题: 问题ISEP 给定矩阵A∈SR~n×n和向量b∈R~n,求实数λ和向量X∈R~n使得 AX=λX+b, (1) ||X||_2=1. (2) 若b=0,则问题ISEP就是通常的实对称矩阵特征值问题,若b≠0,则问题ISEP称为非齐次对称特征值问题,使(1)和(2)式成立的数λ和向量X分别称为非齐次特征值和相应的非齐     

A STOPPING CRITERION FOR HIGHER-ORDER SWEEPING SCHEMES FOR STATIC HAMILTON-JACOBI EQUATIONS

We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme for static Hamilton-Jacobi equations with convex Hamiltonians and present numerical examples that validate the effectiveness of the new stopping criterion.     

非均匀介质中波动方程的部分不变解

讨论了来自于非均匀介质中波动方程的部分不变解的存在性，证明了在波速满足适当的条件下部分不变解是存在的，并得到了部分不变解。     

A NOTE ON THE NONLINEAR CONJUGATE GRADIENT METHOD

The conjugate gradient method for unconstrained optimization problems varies with a scalar. In this note, a general condition concerning the scalar is given, which ensures the global convergence of the method in the case of strong Wolfe line searches. It is also discussed how to use the result to obtain the convergence of the famous Fletcher-Reeves, and Polak-Ribiere-Polyak conjugate gradient methods. That the condition cannot be relaxed in some sense is mentioned.