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Proceedings of the American Mathematical Society

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



Time-variable singularities for solutions of the heat equation

Author: D. V. Widder
Journal: Proc. Amer. Math. Soc. 32 (1972), 209-214
MSC: Primary 35K05; Secondary 44A15
MathSciNet review: 0294906
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Abstract: A solution $ u(x,t)$ of the two-dimensional heat equation $ {u_{xx}} = {u_t}$ may have the representation

$\displaystyle u(x,t) = \int_{ - \infty }^\infty {k(x - y,t)\;d\alpha (y)} $

where $ k(x,t) = {(4\pi t)^{ - 1/2}}\exp [ - {x^2}/(4t)]$, valid in some strip $ 0 < t < c$ of the x, t-plane. If so, $ u({x_0},t)$ is known to be an analytic function of the complex variable t in the disc $ \operatorname{Re} (1/t) > 1/c$, for each fixed real $ {x_0}$. It is shown here that if $ \alpha (y)$ is nondecreasing and not absolutely continuous then $ u({x_0},t)$ must have a singularity at $ t = 0$. Examples show that both restrictions on $ \alpha (y)$ are necessary for that conclusion. It is shown further under the same hypothesis on $ \alpha (y)$, that for each fixed positive $ {t_0} < c,u(x,{t_0})$ is an entire function of x of order 2 and of type $ 1/(4{t_0})$. Compare the function $ k(x,t)$ itself for a check on both conclusions.

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Keywords: Heat equation, fundamental solution, Gaussian integral, entire function, order and type of entire function, singularity of analytic function
Article copyright: © Copyright 1972 American Mathematical Society