Bounds for the fundamental solution of a parabolic equation

Abstract

possesses a fundamental solution provided that the coefficients are Holder continuous. Here x— (xi, • • • , xn) denotes a point in E n with n E= 1, t denotes a point on the real line, and we employ the convention of summation over repeated indices. The fundamental solution g(x, t; £, r) can be constructed by the classical parametrix method, and it satisfies the inequality O^g^Ky, where y is the fundamental solution of aAu = Ut for some constant a>0 and K>0 is a constant which depends upon the Holder norms of the coefficients ([4], [5]). Several authors have investigated the problem of bounding g from below. Il'in, Kalashnikov, and Oleinik [5] proved that gg^const (t—r)~ in the paraboloid |#—£| 2 ^const ( /—r) ; while Besala [3] and Friedman [4] have derived lower bounds for g which are valid when t—r is bounded away from zero. In the appendix to his important paper [6] on Holder continuity of solutions of parabolic and elliptic equations, Nash asserts the existence of global upper and lower bounds for the fundamental solution of the divergence structure parabolic equation