The explicit solution of a diffusion equation with singularity

Watanabe, Michiaki; Watanabe, Shuji

doi:10.1090/S0002-9939-98-04478-5

The explicit solution of a diffusion equation with singularity
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by Michiaki Watanabe and Shuji Watanabe PDF

Proc. Amer. Math. Soc. 126 (1998), 383-389 Request permission

Abstract:

We give the explicit solution of the Cauchy problem for the diffusion equation with a singular term: \[ (\partial / \partial t ) u = ( \partial / \partial x )^2 u - ( k / x^2 ) u \; , \quad t > 0 \; , \quad x \in \mathbf {R}^1 \; ; \] \[ u( 0, x) = f(x) \; , \quad x \in \mathbf {R}^1 \; , \] 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

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
Yoshio Ohnuki and Susumu Kamefuchi, Quantum field theory and parastatistics, University of Tokyo Press, Tokyo; Springer-Verlag, Berlin, 1982. MR 687830, DOI 10.1007/978-3-642-68622-1
Yoshio Ohnuki and Shuji Watanabe, Selfadjointness of the operators in Wigner’s commutation relations, J. Math. Phys. 33 (1992), no. 11, 3653–3665. MR 1185839, DOI 10.1063/1.529860
Michiaki Watanabe and Shuji Watanabe, Selfadjointness of the momentum operator with a singular term, Proc. Amer. Math. Soc. 107 (1989), no. 4, 999–1004. MR 984821, DOI 10.1090/S0002-9939-1989-0984821-8
Shuji Watanabe, Sobolev type theorems for an operator with singularity, Proc. Amer. Math. Soc. 125 (1997), no. 1, 129–136. MR 1343728, DOI 10.1090/S0002-9939-97-03523-5
Shuji Watanabe, An embedding theorem of Sobolev type for an operator with singularity, Proc. Amer. Math. Soc. 125 (1997), no. 3, 839–848. MR 1353406, DOI 10.1090/S0002-9939-97-03642-3
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13