Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The explicit solution of a diffusion equation with singularity

Authors: Michiaki Watanabe and Shuji Watanabe
Journal: Proc. Amer. Math. Soc. 126 (1998), 383-389
MSC (1991): Primary 35K15, 35K22; Secondary 42A38
MathSciNet review: 1459156
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give the explicit solution of the Cauchy problem for the diffusion equation with a singular term:

\begin{displaymath}(\partial / \partial t ) \, u = ( \partial / \partial x )^2 \, u - ( k / x^2 ) \, u \; , \quad t > 0 \; , \quad x \in \mathbf{R}^1 \; ; \end{displaymath}

\begin{displaymath}u( 0, x) = f(x) \; , \quad x \in \mathbf{R}^1 \; , \end{displaymath}

where $k > - 1/4$. We construct the solution on the basis of a generalization of the Fourier transform. We next show that the solution is expressed by an analytic semigroup, and examine smoothness of $x \mapsto u(t, x)$ and continuity of $x \mapsto u(t, x) / x^{\beta}\left( \beta > 0 \right)$.

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

  • [E] A. Erdélyi (ed.), Tables of integral transforms, vol. II, McGraw-Hill, New York, 1954. MR 16:468c
  • [G] J. A. Goldstein, Semigroups of linear operators and applications, Oxford University Press, New York, 1985 / Clarendon Press, Oxford, 1985. MR 87c:47056
  • [OK] Y. Ohnuki and S. Kamefuchi, Quantum field theory and parastatistics, University of Tokyo Press, Tokyo, 1982 / Springer-Verlag, Berlin, Heidelberg and New York, 1982. MR 85b:81001
  • [OW] Y. Ohnuki and S. Watanabe, Self-adjointness of the operators in Wigner's commutation relations, J. Math. Phys. 33 (1992), 3653-3665. MR 93h:81065
  • [WW] M. Watanabe and S. Watanabe, Self-adjointness of the momentum operator with a singular term, Proc. Amer. Math. Soc. 107 (1989), 999-1004. MR 90g:81035
  • [W1] S. Watanabe, Sobolev type theorems for an operator with singularity, Proc. Amer. Math. Soc. 125 (1997), 129-136. MR 97c:47044
  • [W2] S. Watanabe, An embedding theorem of Sobolev type for an operator with singularity, Proc. Amer. Math. Soc. 125 (1997), 839-848. MR 97e:46042
  • [Wi] E. P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations?, Phys. Rev. 77 (1950), 711-712. MR 11:706e
  • [Y] L. M. Yang, A note on the quantum rule of the harmonic oscillator, Phys. Rev. 84 (1951), 788-790. MR 13:804e

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35K15, 35K22, 42A38

Retrieve articles in all journals with MSC (1991): 35K15, 35K22, 42A38

Additional Information

Michiaki Watanabe
Affiliation: Faculty of Engineering, Niigata University, Niigata 950-21, Japan

Shuji Watanabe
Affiliation: Department of Mathematics, Toyota National College of Technology, Eisei-Cho 2-1, Toyota-Shi 471, Japan

Keywords: Diffusion equation with singularity, generalized Fourier transform, analytic semigroup.
Received by editor(s): May 7, 1996
Additional Notes: The second author was partially supported by Grant-in-Aid for Scientific Research (No.07740175), Ministry of Education, Science, Sports and Culture.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society