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Uniqueness in the Cauchy problem for the heat equation

Published online by Cambridge University Press:  20 January 2009

Soon-Yeong Chung
Soon-Yeong Chung
Affiliation:
Department of Mathematics, Sogang University, Seoul 121–742, Korea, E-mail address: sychung@ccs.sogang.ac.kr
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We relax the growth condition in time for uniqueness of solutions of the Cauchy problem for the heat equation as follows: Let u(x, t) be a continuous function on ℝn × [0, T] satisfying the heat equation in ℝn × (0, t) and the following:

(i) There exist constants a > 0, 0 < α < 1, and C > 0 such that

(ii) u(x, 0) = 0 for x ∈ ℝn.

Then u(x, t)≡ 0 on ℝn × [0, T]

We also prove that the condition 0 < α < 1 is optimal.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 455 - 468
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

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