HARNACK INEQUALITY AND GAUSSIAN BOUNDS FOR PARABOLIC EQUATIONS WITH SINGULAR LOWER-ORDER COEFFICIENTS

Abstract: We study the evolution family generated by second-order parabolic operators of the form

∂_tu − ∇ · (a(t, x)∇u) + b(t, x) · ∇u − ∇ · (b̃(t, x)u) + V(t, x)u.

The diffusion matrix a is uniformly elliptic and Dini continuous in the spatial variable. The lower-order coefficients b, b̃, V belong to the parabolic Kato classes K_l+1 and K_l+2, which are optimal for perturbation theory. Using Moser’s iteration we prove the Harnack inequality and then the two-sided Gaussian bound for the fundamental solution:

c (t − s)^−l/2 exp(−|x − y|² / (C(t − s))) ≤ p(t, x; s, y) ≤ C (t − s)^−l/2 exp(−|x − y|² / (c(t − s))),

with constants depending only on the ellipticity bounds and the Kato norms. As applications we obtain well-posedness of the Cauchy problem in L¹ ∩ L^∞ and a weighted maximum principle.

Keywords: Harnack inequality, Gaussian bounds, parabolic equations, fundamental solution, singular potentials, Kato class, Dini continuity, Moser iteration, perturbation theory, heat kernel.

MSC: 35K08, 35K10, 31B35, 35J10, 35B45, 35D30, 47D07.

DOI 10.56082/annalsarscimath.2026.4.37

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^*Accepted for publication on July 01, 2026
^†math.kiev@gmail.com, National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute Kyiv, Ukraine

PUBLISHED in

Annals Academy of Romanian Scientists Series on Mathematics and Its Application,

Volume 18 no 4, 2026

ISSN ONLINE 2066 – 6594
ISSN PRINT 2066 – 5997