The linear heat equation with highly oscillating potential

Kombe, Ismail

doi:10.1090/S0002-9939-04-07392-7

The linear heat equation with highly oscillating potential
HTML articles powered by AMS MathViewer

by Ismail Kombe

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

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

Published electronically: April 9, 2004

PDF | Request permission

Abstract:

In this paper we consider the following initial value problem: \[ \begin {cases} \frac {\partial u}{\partial t}=-Hu+V(x)u & \text {in $\mathbb {R}^N\times (0,T)$},\\ u(x,0) = u_0 (x)\geq 0 & \text {on $\mathbb {R}^N \times \{t=0\}$}, \end {cases} \] where $H=-\Delta -\frac {\beta }{|x|^2}\sin (\frac {1}{|x|^{\alpha }})$ and $0\le V\in L_{\text {loc}}^1(\mathbb {R}^N)$. Nonexistence of positive solutions is analyzed.

References

J. A. Aguilar Crespo and I. Peral Alonso, Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Anal. 31 (2000), no. 6, 1270–1294. MR 1766560, DOI 10.1137/S0036141098341137
Pierre Baras and Jerome A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), no. 1, 121–139. MR 742415, DOI 10.1090/S0002-9947-1984-0742415-3
Xavier Cabré and Yvan 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), no. 11, 973–978 (French, with English and French summaries). MR 1733904, DOI 10.1016/S0764-4442(00)88588-2
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), no. 2, 441–476. MR 1616905, DOI 10.1006/jdeq.1997.3375
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.
J. A. Goldstein and I. Kombe, Nonlinear degenerate parabolic equations with singular lower-order term, Advances in Differential Equations 8 (2003), 1153–1192.
Jerome A. Goldstein and Qi S. Zhang, On a degenerate heat equation with a singular potential, J. Funct. Anal. 186 (2001), no. 2, 342–359. MR 1864826, DOI 10.1006/jfan.2001.3792
Jerome A. Goldstein and Qi S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc. 355 (2003), no. 1, 197–211. MR 1928085, DOI 10.1090/S0002-9947-02-03057-X
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
Karl-Theodor Sturm, Schrödinger operators with highly singular, oscillating potentials, Manuscripta Math. 76 (1992), no. 3-4, 367–395. MR 1185026, DOI 10.1007/BF02567767