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ESTIMATES FOR FUNDAMENTAL SOLUTIONS OF SECOND-ORDER PARABOLIC EQUATIONS

Published online by Cambridge University Press:  08 January 2001

VITALI LISKEVICH  and
YULI SEMENOV
VITALI LISKEVICH
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW; [email protected]
YULI SEMENOV
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3; [email protected]
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Abstract

In this paper we study the second-order parabolic equation

formula here

in a domain [0,T]×ℝd ⊂ ℝd+1, where a = (aij)di,j=1 is matrix of bounded measurable coefficients, b = (bj)dj=1, and bˆ = (bˆj)dj=1 are measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and ∂tu = ∂u/∂t, and we employ the notation

formula here

We introduce a new class of coefficients in the lower-order terms for which we prove the existence and the uniqueness of the weak fundamental solution, and for this we derive Gaussian upper and lower bounds.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 2 , October 2000 , pp. 521 - 543
Copyright
The London Mathematical Society 2000

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