The rate of spatial decay of non-negative solutions of linear parabolic equations

Abstract

for all xelR", then u = 0 on IR"x [0, t0]. Here to>0, X = ( X 1 . . . . . X n ) ~ n and [Ixl[ = 1/(x~ + . . + x.~). This question was taken up by GUSAROV in [4] and [5; Theorem 6]. He established the truth of the conjecture, and in fact showed that if u(x,t) =O(exp (kllxII2)), then there is a constant C, depending upon to, such that if u(x, to)= O ( e x p ( C ]]xll2)), then u=0. We show below that if the growth condition u(x,t)=O(exp(kHxll2)) is replaced by the assumption that u > 0, then a result similar to that of GUSAROV can be established for a very general class of linear second order parabolic equations, with coefficients depending upon t as well as x. This change in the hypothesis on u parallels that made by BOCHER [3] when dealing with elliptic equations, and the results given below are very closely related to the analogues of B0CHER's theorem presented in [1] (as well as in some of the works cited therein). We are concerned here with non-negative (weak) solutions of linear parabolic equations on a strip IR" x ]0, to]. We require that each such solution u has a representation in the form