Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The heat equation with a singular potential


Authors: Pierre Baras and Jerome A. Goldstein
Journal: Trans. Amer. Math. Soc. 284 (1984), 121-139
MSC: Primary 35K05; Secondary 60J65
MathSciNet review: 742415
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Of concern is the singular problem $ \partial u/\partial t = \Delta u + (c/\vert x{\vert^2})\,u + f(t,x), u(x,0) = u_{0}(x)$, and its generalizations. Here $ c \geqslant 0,x \in {{\mathbf{R}}^N},t > 0$, and $ f$ and $ {u_0}$ are nonnegative and not both identically zero. There is a dimension dependent constant $ {C_{\ast} }(N)$ such that the problem has no solution for $ c > {C_{\ast} }(N)$. For $ c \leqslant {C_{\ast} }(N)$ necessary and sufficient conditions are found for $ f$ and $ {u_0}$ so that a nonnegative solution exists.


References [Enhancements On Off] (What's this?)

  • [1] P. Baras, in preparation.
  • [2] Pierre Baras and Jerome A. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential equations (Birmingham, Ala., 1983) North-Holland Math. Stud., vol. 92, North-Holland, Amsterdam, 1984, pp. 31–35. MR 799330, 10.1016/S0304-0208(08)73675-2
  • [3] H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
  • [4] Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 0170091
  • [5] Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 0159138
  • [6] S. I. Rosencrans, Diffusions and partial differential equations, Lecture Notes, Tulane Univ., New Orleans, 1977-78.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K05, 60J65

Retrieve articles in all journals with MSC: 35K05, 60J65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0742415-3
Keywords: Heat equation, singular potential, inverse square potential, nonexistence, Feynman-Kac formula
Article copyright: © Copyright 1984 American Mathematical Society