ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 47, 35 Xu, Hong-Kun, University of Durban-Westville, South Africa Asymptotic behavior of a gradient flow It is known that if φ is a convex and even function on a Hilbert space H such that min_{H} φ is achieved, then the (weak) solution u of the initial value problem du/dt + ∂φ(u) ∋ 0, u(0) = x converges strongly as t → ∞. However, it is unknown whether the same is true of the solution u to the inhomogeneous problem du/dt+∂φ(u) ∋ f, u(0) = x, where f ∈ H. (This is a question asked by H. Brezis.) The main purpose of this paper is to give a partial answer to this question. Reference: H. Brezis, Asymptotic behavior of some evolution systems, in “Nonlinear Evolution Equations” (M. G. Crandall, Ed.), Academic Press, 1978, pp. 141–154.