Boundary behavior of positive solutions of the heat equation on a semi-infinite slab

Mair, B. A.

doi:10.1090/S0002-9947-1986-0833703-2

Boundary behavior of positive solutions of the heat equation on a semi-infinite slab
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Trans. Amer. Math. Soc. 295 (1986), 687-697 Request permission

Abstract:

In this paper, the abstract Fatou-Naim-Doob theorem is used to investigate the boundary behavior of positive solutions of the heat equation on the semi-infinite slab $X = {{\mathbf {R}}^{n - 1}} \times {{\mathbf {R}}_ + } \times (0,T)$. The concept of semifine limit is introduced, and relationships are obtained between fine, semifine, parabolic, one-sided parabolic and two-sided parabolic limits at points on the parabolic boundary of $X$. A Carleson-Calderón-type local Fatou theorem is also obtained for solutions on a union of two-sided parabolic regions.

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