Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The linear heat equation with highly oscillating potential

Author(s): Ismail Kombe
Journal: Proc. Amer. Math. Soc. 132 (2004), 2683-2691.
MSC (2000): Primary 35K15, 35K25, 35R25
Posted: April 9, 2004
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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

\begin{displaymath}\begin{cases} \frac{\partial u}{\partial t}=-Hu+V(x)u & \text... ...)\geq 0 & \text{on}\quad\mathbb{R}^N \times\{t=0\}, \end{cases}\end{displaymath}

where $ H=-\Delta -\frac{\beta}{\vert x\vert^2}\sin(\frac{1}{\vert x\vert^{\alpha}}) $ and $0\le V\in L_{\text{loc}}^1(\mathbb{R}^N)$. Nonexistence of positive solutions is analyzed.

References:

[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

Similar Articles:

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: 10.1090/S0002-9939-04-07392-7
PII: S 0002-9939(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
Posted: April 9, 2004
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2004, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google