The DeGiorgi–Nash–Moser type of estimate for parabolic Volterra integrodifferential equations

Bei Hu and Hong-Ming Yin

Vol. 178 (1997), No. 2, 265–277

DOI: 10.2140/pjm.1997.178.265

Abstract

The DeGiorgi-Nash-Moser estimate plays a crucial role in the study of quasilinear elliptic and parabolic equations. In the present paper we shall show that this fundamental estimate holds for solutions of a linear parabolic Volterra integrodifferential equation:

where {a_ij} and {b_ij} are only assumed to be measurable, bounded and {a_ij} satisfy a strong ellipticity condition. The proof is based on ℒ^2,μ theory for parabolic equations. A global solvability result in the classical sense for a class of quasilinear parabolic integrodifferential equations is presented as an application of the general results.

Milestones

Received: 15 June 1995

Revised: 10 November 1995

Published: 1 April 1997