GAUSSIAN BOUNDS FOR THE FUNDAMENTAL-SOLUTIONS OF DEL(A-DEL-U)-DEL-U-U(T)=0(B)

We study the parabolic equation del(A del u) + B del u - u(t) = 0, whe re B = B(x, t) is in a class of singular functions more general than t he L-p,L-q class in Aronson's paper [1]. We prove the existence of Gau ssian bounds for the fundamental solutions. As a corollary we show tha t if \B\ is an element of K-n,K-1 and B = B(x) is compactly supported, then the heat kernel of Delta + B del has Gaussian upper aad lower bo und. In the paper [8]. B. Simon obtained Gaussian bounds when the zero order term is singular, However the case of singular drift terms (cor responding to magnetic fields) has been open. This question is settled by the corollary, In addition we give a condition so that the upper b ound is global in time.