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

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

MathSciNet review:
1360232

Full-text PDF

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*, Annali della Scuola Norm. Sup. Pisa**XXII**(1968), 607-694. MR**55:8553****[AS]**M. Aizenman and B. Simon,*Brownian motion and Harnack's inequality for Schrödinger operators*, Comm. Pure Appl. Math**35**(1982), 209-271. MR**84a:35062****[CFG]**F. Chiarenza, E. Fabes and N. Garofalo,*Harnack's inequality for Schrödinger operators and the continuity of solutions*, Proc. Amer. Math. Soc.**307**(1986), 415-425. MR**88a:35037****[CZ]**M. Cranston and Z. Zhao,*Conditional transformation of drift formula and potential theory for*, Comm. Math. Phys.**112**(1987), 613-625. MR**89g:35030****[CFZ]**M.Cranston, E.Fabes, Z.Zhao,*Conditional gauge and potential theory for the Schrödinger operator*, Trans. Amer. Math. Soc.**307**(1988), 171-194. MR**90a:60135****[FS1]**E.B. Fabes and D.W. Stroock,*The -integrability of Green's functions and fundamental solutions for elliptic and parabolic equations*, Duke Math. J.**51**(1984), 997-1016. MR**86g:35057****[FS2]**E.B. Fabes and D.W. Stroock,*A new proof of Moser's parabolic Harnack inequality using the old idea of Nash*, Arch. Rational Mech. Anal.**96**(1986), 327-338. MR**88b:35037****[GW]**M. Gruter and K.O. Widman,*The Green function for uniformly elliptic equations*, Manu. Math.**37**(1982), 303-342. MR**83h:35033****[S]**K. Sturm,*Harnack's inequality for parabolic operators with singular low order terms*, Math. Z.**216**(1994), 593-612. MR**95g:35027****[T]**F. Treves,*Basic Linear Partial Differential Equations*, Academic Press, 1975. MR**56:6063**

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