Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-15T09:56:59.440Z Has data issue: false hasContentIssue false
  • English
  • Français

A NON-LINEAR EVOLUTION EQUATION DRIVEN BY LOGARITHMIC POTENTIAL

Published online by Cambridge University Press:  01 May 2000

KÔHEI UCHIYAMA
KÔHEI UCHIYAMA
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan
Get access

Abstract

We study the behaviour of non-negative solutions to the equation

formula here

This equation is an ‘explicitly’ soluble special case of a class of non-local evolution equations, of which the behaviour of solutions has been studied by Uchiyama [9]. In this paper, certain fine properties of solutions such as a local relaxation within their supports are obtained for the present special case. The asymptotic forms (for large time) of the solutions whose initial measures have compact supports are identified within the error of the order O(t−3/2). It is also shown that the comparison principle does not hold. With a simple change of variables, the equation ([midast ]) is transformed to the equation arising in Dyson's approach to the Wigner semi-circle law of eigenvalues of random matrices, and the present results have immediate consequences on the latter.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 32 , Issue 3 , May 2000 , pp. 353 - 363
Copyright
© The London Mathematical Society 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)