Topic
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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01 Jan 1985
6 citations
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TL;DR: In this paper, the authors considered the non-local parabolic problem with a homogeneous Dirichlet boundary condition, where λ > 0, p > 0 and f is nonincreasing.
Abstract: In this paper, we consider the asymptotic behaviour for the non-local parabolic problem \\begin{eqnarray} \\[ u_{t}=\\Delta u+\\displaystyle\\frac{\\lambda f(u)}{(\\int_{\\Omega}f(u)dx)^{p}},\\quad x\\in \\Omega,\\ t>0, \\\\end{eqnarray} with a homogeneous Dirichlet boundary condition, where λ > 0, p > 0 and f is non-increasing. It is found that (a) for 0 < p ≤ 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 < p < 2, u(x, t) is globally bounded for any λ > 0; (c) for p = 2, if 0 < λ < 2|∂Ω|2, then u(x, t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution and u(x, t) is a global solution and u(x, t) → ∞ as t → ∞ for all x ∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution and u(x, t) blows up in finite time for all x ∈ Ω; (d) for p > 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* and u0(x) sufficiently large, u(x, t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour of u(x, t) as it blows up are obtained for p ≥ 2.
6 citations
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TL;DR: In this paper, the authors generalize LeCam's third lemma by using the rate of convergence in the case of asymptotically efficient test statistics, which is specified to linear combinations of order statistics and linear rank statistics.
Abstract: Asymptotic distributions of test statistics under alternatives are important from the point of view of their power properties. When the limiting distributions of test statistics are specified under the hypothesis in a certain sense, LeCam's third lemma ([4], Chapter 6) enables one to obtain their limiting distributions under close alternatives. In this paper we generalize LeCam's third lemma by using the rate of convergence in the case of asymptotically efficient test statistics. A general lemma is proved which is specified to linear combinations of order statistics (L-statistics) and linear rank statistics (R-statistics). Edgeworth-type asymptotic expansions for these statistics under alternatives are considered in [3].
6 citations
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TL;DR: The principal part of an asymptotic expansion at infinity of the logarithm of integral functions of finite order with simple positive zeros was determined in this paper, with the aid of a Cauchy-type integral with smooth density.
Abstract: The principal part of an asymptotic expansion at infinity of the logarithm of integral functions of finite order with simple positive zeros is determined. The asymptotic form is obtained with the aid of a Cauchy-type integral with smooth density.
6 citations