On A parabolic equation with a singular lower order term

Author:
Qi Zhang

Journal:
Trans. Amer. Math. Soc. **348** (1996), 2811-2844

MSC (1991):
Primary 35K10

MathSciNet review:
1360232

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain the existence of the weak Green's functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green's function, we prove a Harnack inequality for the non-negative weak solutions.

**[A]**D. G. Aronson,*Non-negative solutions of linear parabolic equations*, Ann. Scuola Norm. Sup. Pisa (3)**22**(1968), 607–694. MR**0435594****[AS]**M. Aizenman and B. Simon,*Brownian motion and Harnack inequality for Schrödinger operators*, Comm. Pure Appl. Math.**35**(1982), no. 2, 209–273. MR**644024**, 10.1002/cpa.3160350206**[CFG]**F. Chiarenza, E. Fabes, and N. Garofalo,*Harnack’s inequality for Schrödinger operators and the continuity of solutions*, Proc. Amer. Math. Soc.**98**(1986), no. 3, 415–425. MR**857933**, 10.1090/S0002-9939-1986-0857933-4**[CZ]**M. Cranston and Z. Zhao,*Conditional transformation of drift formula and potential theory for 1\over2Δ+𝑏(⋅)⋅∇*, Comm. Math. Phys.**112**(1987), no. 4, 613–625. MR**910581****[CFZ]**M. Cranston, E. Fabes, and Z. Zhao,*Conditional gauge and potential theory for the Schrödinger operator*, Trans. Amer. Math. Soc.**307**(1988), no. 1, 171–194. MR**936811**, 10.1090/S0002-9947-1988-0936811-2**[FS1]**E. B. Fabes and D. W. Stroock,*The 𝐿^{𝑝}-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations*, Duke Math. J.**51**(1984), no. 4, 997–1016. MR**771392**, 10.1215/S0012-7094-84-05145-7**[FS2]**E. B. Fabes and D. W. Stroock,*A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash*, Arch. Rational Mech. Anal.**96**(1986), no. 4, 327–338. MR**855753**, 10.1007/BF00251802**[GW]**Michael Grüter and Kjell-Ove Widman,*The Green function for uniformly elliptic equations*, Manuscripta Math.**37**(1982), no. 3, 303–342. MR**657523**, 10.1007/BF01166225**[S]**Karl-Theodor Sturm,*Harnack’s inequality for parabolic operators with singular low order terms*, Math. Z.**216**(1994), no. 4, 593–611. MR**1288047**, 10.1007/BF02572341**[T]**François Trèves,*Basic linear partial differential equations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 62. MR**0447753**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35K10

Retrieve articles in all journals with MSC (1991): 35K10

Additional Information

**Qi Zhang**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Email:
zhangq@math.purdue.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01675-3

Received by editor(s):
September 26, 1994

Received by editor(s) in revised form:
May 28, 1995

Article copyright:
© Copyright 1996
American Mathematical Society