Extinction Properties of Solutions to Hyperbolic Equations

In this paper the extinction properties of solutions to both linear and nonlinear hyperbolic equations, which reflect the effect of damping boundary conditions and have not been explicitly stated in literature before, are introduced and extensively studied.     

Computing Asymptotic Invariants with the Ricci Tensor on Asymptotically Flat and Asymptotically Hyperbolic Manifolds

We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.     

Maslov's Canonical Operator in Problems on Localized Asymptotic Solutions of Hyperbolic Equations and Systems

An analog of Maslov's canonical operator is defined for functions localized in a neighborhood of subsets of positive codimension.

     

Asymptotic Behaviour of Solutions to some Pseudoparabolic Equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u_t−Δu_t−νΔu−(b, ∇u)=∇⋅F(u), u(x, 0)=u₀(x), where ν>0 is a fixed constant, t⩾0, x∈ℝⁿ. First, we prove that if u is the solution to the linearized equation, i.e. with ∇⋅F(u)≡0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long-time behaviour of u is described by the heat kernel. Next, analogous results are established for the non-linear equation with some assumptions imposed on F, p, and the initial condition u₀. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Studies of Some Curvature Operators in a Neighborhood of an Asymptotically Hyperbolic Einstein Manifold

On an asymptotically hyperbolic Einstein manifold (M,g₀) for which the Yamabe invariant of the conformal structure on the boundary at infinity is nonnegative, we show that the operators of Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the metric g₀. We deduce in the C^∞ case that the image of the Riemann-Christoffel curvature operator is a submanifold in a neighborhood of g₀.     

Existence of Solutions for Asymptotically 'Linear' p-Laplacian Equations

Under very general conditions, a proof is given, via Morse theory,of the existence of solutions for asymptotically ‘linear’p-Laplacian equations, where the asymptotic limit may be greaterthan the second eigenvalue. The existence of nonzero solutionsis also considered. 2000 Mathematics Subject Classification35J65, 58E05.     

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??_Bu + λu = a(z)f(u), u ≥ 0 in R^N₊,where N = n + 1 ≥ 3, λ > 0, z = t,x₁,· · · ,x_n, and ?_B = (t?_t)^{2 + ?2_x1 + · · · + ?2_xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.}     

一类非牛顿流体模型解的渐近性态(英)

本文主要讨论一类带 $p \,\,( 1+\frac{2n}{n+2} \leq p<3 )\,$ 幂增长耗散位势的非牛顿流体模型解的渐近性态, 利用改进的 Fourier分解方法, 证明了其解在$L^2$ 范数下衰减率为 $(1+t)^{-\frac{n}{4}}$.     

一类非牛顿流体模型解的渐近性态

本文主要讨论一类带p(1 2n/(n 2)■p＜3)幂增长耗散位势的非牛顿流体模型解的渐近性态,利用改进的Fourier分解方法,证明了其解在L~2范数下衰减率为(1 t)~(-n/4)．     

Asymptotic Behavior of Solutions to Second-Order Nonlinear Differential Equations

In this paper the authors study the oscillation and the asymptotic behavior of solutiosn to the second-order nonlinear differential equations $[\ddot x]\pmX(t,x,[\dot x])=0$(X_{\pm}) and give necessary and sufficient conditions for Equation (X_+) to have a bounded nonoscillatory solution or to be oscillatory. There are four classes of solutions to Equation (X_) with different asymptotic behavior. For each class of solution, the necessary or sufficient condition of the existence is obtained.     

三维Wigner-Poisson方程解的渐近性

考虑三维ＷＰ方程的Ｃａｕｃｈｙ问题,在吸引力场的情形下,证明了解的拟渐近性．     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

On Global Classical Solutions of Hyperbolic Differential-Difference Equations

Doklady Mathematics - A one-parameter family of global solutions of a two-dimensional hyperbolic differential-difference equation with an operator acting with respect to a space variable is...     

Asymptotic Formulas for the Solutions of Integro-Differential Equations

We study the problem of asymptotic integration of the linear integro-differential equation

Applications of Fixed Point Theorems to the Vacuum Einstein Constraint Equations with Non-Constant Mean Curvature

In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer’s fixed point theorem (known methods use Schauder’s fixed point theorem), while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These methods allow to: unify some recent existence results, simplify many proofs (for instance, the one of the main theorems in Dahl et al., Duke Math J 161(14):2669–2697, 2012) and weaken the assumptions of many recent results.     

Classical Solutions of Hyperbolic Equations with Nonlocal Potentials

Doklady Mathematics - A three-parameter family of global solutions for a two-dimensional hyperbolic differential-difference equation with a nonlocal potential is constructed. A theorem that the...