The linear heat equation with highly oscillating potential

Author:
Ismail Kombe

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2683-2691

MSC (2000):
Primary 35K15, 35K25, 35R25

DOI:
https://doi.org/10.1090/S0002-9939-04-07392-7

Published electronically:
April 9, 2004

MathSciNet review:
2054795

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the following initial value problem:

where and . Nonexistence of positive solutions is analyzed.

**[1]**J. A. Aguilar Crespo and I. Peral Alonso,*Global behaviour of the Cauchy problem for some critical nonlinear parabolic equations*, SIAM J. Math. Anal.,**31**(2000), 1270-1294. MR**2001d:35086****[2]**P. Baras and J. A. Goldstein,*The heat equation with a singular potential*, Trans. Amer. Math. Soc.**284**(1984), 121-139.MR**85f:35099****[3]**X. Cabré and Y. Martel,*Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier*, C. R. Acad. Sci. Paris Sér. I. Math.,**329**(1999), 973-978. MR**2000j:35117****[4]**L. C. Evans,*Partial Differential Equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, Rhode Island, 1998. MR**99e:35001****[5]**J. Garcia Azorero and I. Peral Alonso,*Hardy inequalities and some critical elliptic and parabolic problems*, J. Differential Equations,**144**(1998), 441-476. MR**99f:35099****[6]**J. A. Goldstein and Ismail Kombe,*Instantaneous blow up*, Advances in differential equations and mathematical physics (Birmingham, AL, 2002), Contemp. Math., Vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 141-150.**[7]**J. A. Goldstein and I. Kombe,*Nonlinear degenerate parabolic equations with singular lower-order term*, Advances in Differential Equations**8**(2003), 1153-1192.**[8]**J. A. Goldstein and Q. S. Zhang,*On a degenerate heat equation with a singular potential*, J. Functional Analysis,**186**(2001), 342-359. MR**2002k:35179****[9]**J. A. Goldstein and Q. S. Zhang,*Linear parabolic equations with strong singular potentials*, Trans. Amer. Math. Soc.**355**(2003), 197-211. MR**2003h:35096****[10]**G. M. Lieberman,*Second Order Parabolic Differential Equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.MR**98k:35003****[11]**J. Moser,*A Harnack inequality for parabolic differential equations*, Comm. Pure Appl. Math.**17**(1964), 101-134. MR**28:2357****[12]**K.-T. Sturm,*Schrödinger operators with highly singular, oscillating potentials*, Manuscripta Math.**76**(1992), 367-395. MR**94m:35077**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35K15,
35K25,
35R25

Retrieve articles in all journals with MSC (2000): 35K15, 35K25, 35R25

Additional Information

**Ismail Kombe**

Affiliation:
Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211

Email:
kombe@math.missouri.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07392-7

Keywords:
Heat equation,
instantaneous blow up,
positive solutions

Received by editor(s):
April 21, 2003

Received by editor(s) in revised form:
June 18, 2003

Published electronically:
April 9, 2004

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society