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Proceedings of the American Mathematical Society
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
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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 $x \mapsto u(t, x) / x^{\beta}\left( \beta > 0 \right)$.


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Additional Information

Michiaki Watanabe
Affiliation: Faculty of Engineering, Niigata University, Niigata 950-21, Japan
Email: m.watanabe@geb.ge.niigata-u.ac.jp

Shuji Watanabe
Affiliation: Department of Mathematics, Toyota National College of Technology, Eisei-Cho 2-1, Toyota-Shi 471, Japan
Email: swtnb@tctcc.cc.toyota-ct.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04478-5
PII: S 0002-9939(98)04478-5
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