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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
Published electronically: April 9, 2004
MathSciNet review: 2054795
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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.

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  • [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

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Additional Information

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

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

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