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Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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TL;DR: In this paper, the authors studied the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered).
Abstract: We study the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞, where y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function. Introduction The phenomenon called stabilization is well known for parabolic equations both in linear (see e.g. [1] and references therein) and non-linear (see e.g. [2] and references therein) cases; it means the existence of a finite limit of the solution as t → ∞. However, there are well-posed non-isotropic elliptic boundary-value problems in unbounded domains (see e.g. [3]) for which we can talk about stabilization in the following sense: the solution has a finite limit as a selected spatial variable tends to infinity. This paper is devoted to the Dirichlet problem in half-space for elliptic equations. We present necessary and sufficient conditions for the stabilization of its solution; here the spatial variable, orthogonal to the hyperplane of boundary-value data, plays the role of time. In Section 1, the linear case is presented; Sections 2 and 3 are devoted to quasi-linear equations with the so-called Burgers-Kardar-ParisiZhang non-linearity type (see e.g. [4], [5]). Equations with such non-linearities arise, for example, in modeling of directed polymers and interface growth. They also present an independent theoretical interest because they contain second powers of the first derivatives (see e.g. [6] and references therein). Note that we deal with the stabilization problem in cylindrical domains with an unbounded base (in particular, here the base of the cylinder is the whole E ). As in the parabolic case, this problem is principally different (this refers both to the results and to the methods of research) from the stabilization problem in cylindrical domains with a bounded base. The latter problem has been investigated Received by the editors March 6, 2002. 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B40, 35J60.
13 citations
01 Jan 2005
TL;DR: In this paper, the global almost sure asymptotic stability of the trivial solution of nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in IR1.
Abstract: Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by independent (but not necessarily identically distributed) random variables is proven under appropriate conditions in IR1. This result can be used to verify asymptotic stability of stochastic-numerical methods such as partially drift-implicit trapezoidal methods for nonlinear stochastic differential equations with variable step sizes.
13 citations
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01 Jan 1997
TL;DR: The quantum propagation of N-body systems is asymptotically constrained to Lagrangian manifolds corresponding to particular solutions of the free Hamilton-Jacobi equation as mentioned in this paper.
Abstract: The quantum propagation of N-body systems is asymptotically constrained to Lagrangian manifolds corresponding to particular solutions of the free Hamilton-Jacobi equation. This is used to give a proof of asymptotic completeness for short-range interactions.
13 citations
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13 citations
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13 citations