The behavior of the heat operator

on weighted Sobolev spaces

Authors:
G. N. Hile and C. P. Mawata

Journal:
Trans. Amer. Math. Soc. **350** (1998), 1407-1428

MSC (1991):
Primary 35B45; Secondary 35A05, 35J60

DOI:
https://doi.org/10.1090/S0002-9947-98-02140-0

MathSciNet review:
1467466

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Denoting by the heat operator in , we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which is an injection, a surjection, or an isomorphism. We also describe the range and kernel of in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in .

**1.**H. Begehr and G. N. Hile,*Schauder estimates and existence theory for entire solutions of linear elliptic equations*, Proc. of Royal Soc. Edinburgh**110A**(1988), 101-123. MR**90a:35004****2.**M. Cantor,*Spaces of functions with asymptotic conditions on*, Indiana Univ. Math. J.**24**(1975), 897-902. MR**51:1873****3.**A. Friedman,*Partial Differential Equations of Parabolic Type*, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR**31:6062****4.**D. Gilbarg and N. S. Trudinger,*Elliptic Partial Differential Equations of Second Order*, 2nd ed., Springer-Verlag, Berlin, 1983. MR**86c:35035****5.**G. N. Hile and C. P. Mawata,*Liouville theorems for nonlinear elliptic equations of second order*, Partial Differential Equations with Real Analysis, Pitman Research Notes in Math. (H. Begehr and A Jeffrey, eds.)**263**(1992), 57-92. MR**93m:35016****6.**G. N. Hile and C. P. Mawata,*Liouville theorems for nonlinear parabolic equations of second order*, Differential and Integral Equations**9**(1) (1996), 149-172. MR**96k:35085****7.**T. Kato,*Perturbation Theory for Linear Operators*, 2nd ed., Springer-Verlag, Berlin, 1976. MR**53:11389****8.**R. B. Lockhart,*Fredholm properties of a class of elliptic operators on non-compact manifolds*, Duke Math. J.**48**(1981), 289-312. MR**82j:35050****9.**R. B. Lockhart and R. C. McOwen,*On elliptic systems in*, Acta Math.**150**(1983), 125-135;**153**(1984), 303-304. MR**84d: 35048**; MR**86a:35049****10.**C. P. Mawata,*Schauder estimates and existence theory for entire solutions of linear parabolic equations*, Differential and Integral Equations**2**(3) (1989), 251-274. MR**90i:35024****11.**R. C. McOwen,*Behavior of the Laplacian on weighted Sobolev spaces*, Comm. Pure Appl. Math.**32**(1979), 783-795. MR**81m:47069****12.**R. C. McOwen,*On elliptic operators in*, Comm. Partial Differential Equations**5**(1980), 913-933. MR**84k:35058****13.**L. Nirenberg and H. F. Walker,*The null spaces of elliptic partial differential operators in*, J. Math. Anal. Appl.**42**(1973), 271-301. MR**47:9354****14.**H. F. Walker,*On the null-spaces of first order elliptic partial differential operators in*, Proc. Amer. Math. Soc.**30**(1971), 278-286. MR**43:6583****15.**H. F. Walker,*On the null-spaces of elliptic partial differential operators in*, Trans. Amer. Math. Soc.**173**(1972), 263-275. MR**46:7972****16.**D. V. Widder,*The Heat Equation*, Academic Press, New York, San Francisco, London, 1975. MR**57:6840****17.**D. V. Widder,*Series expansions of solutions of the heat equation in dimensions*, Ann. Mat. Pura Appl.**55**(1961), 389-409. MR**25:331**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35B45,
35A05,
35J60

Retrieve articles in all journals with MSC (1991): 35B45, 35A05, 35J60

Additional Information

**G. N. Hile**

Affiliation:
University of Hawaii, Manoa, 2565 The Mall, Honolulu, Hawaii 96822

Email:
hile@math.hawaii.edu

**C. P. Mawata**

Affiliation:
University of Tennessee, Chattanooga, 615 McCallie Avenue, Chattanooga, Tennessee 37403

Email:
cmawata@cecasun.utc.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02140-0

Received by editor(s):
April 12, 1996

Article copyright:
© Copyright 1998
American Mathematical Society