A Non-Linear Evolution Equation Driven By Logarithmic Potential

Abstract

We study the behaviour of non-negative solutions to the equation ∂ ∂ t ρ ( x , t ) = ∂ ∂ x ( ρ ( x , t ) ∫ − ∞ ∞ ρ ( y , t ) d y y − x ) , t > 0 , − ∞ < x < ∞ . 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 (*) 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. 2000 Mathematics Subject Classification 45K05, 45G05.