Uniqueness for the Cauchy problem for degenerate parabolic equations

Avner Friedman

Vol. 46 (1973), No. 1, 131–147

DOI: 10.2140/pjm.1973.46.131

Abstract

Consider a second order degenerate parabolic operator L. The present paper is concerned with the uniqueness of solutions of the Cauchy problem: Lu = f in a strip 0 < t ≦ T, u(0,x) = ϕ(x) for all x in Rⁿ. It is proved that there is at most one solution subject to a growth condition which depends on the degeneracy of L. In the special case where L is ultraparabolic, uniqueness is proved under only onesided growth condition. The methods used involve the construction of comparison functions in suitable sequences of domains.

Mathematical Subject Classification 2000

Primary: 35K15

Milestones

Received: 18 February 1972

Published: 1 May 1973