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

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

DOI: https://doi.org/10.1090/S0002-9939-98-04478-5

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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 $x \mapsto u(t, x) / x^{\beta }\left ( \beta > 0 \right )$.

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