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Journal of the London Mathematical Society 2000 62(2):521-543; doi:10.1112/S0024610700001332
© 2000 by London Mathematical Society
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© The London Mathematical Society

Estimates for Fundamental Solutions of Second-Order Parabolic Equations

Vitali Liskevich and Yuli Semenov

School of Mathematics, University of Bristol Bristol BS8 1TW, V.Liskevich{at}bristol.ac.uk
Department of Mathematics, University of Toronto Toronto, Canada M5S 3G3, semenov{at}math.toronto.edu

Received 3 June 1999. Revision received 20 September 1999.

In this paper we study the second-order parabolic equation


Formula

in a domain [0,T]xRd sub Rd+1, where Formula is matrix of bounded measurable coefficients, Formula, and Formula are measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and {partial}tu = {partial}u/{partial}t, and we employ the notation


Formula

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.


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