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A Noncharacteristic Cauchy Problem for Linear Parabolic Equations I: Solvability
Author(s) -
Haò Dinh Nho
Publication year - 1995
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19951710112
Subject(s) - mathematics , cauchy problem , divergence (linguistics) , mathematical analysis , initial value problem , constant (computer programming) , class (philosophy) , cauchy distribution , pure mathematics , linguistics , philosophy , artificial intelligence , computer science , programming language
Noncharacteristic Cauchy problems for parabolic equations arc frequently encountered in many areas of heat transfer. These problems are well known to be severely ill‐posed. In this paper a solvability criterion for a class of such problems is established. It is proved that a weak solution of a noncharacteristic Cauchy problem for linear parabolic equations in divergence form with coefficients in a Holmgren class 2 in time exists if and only if the Cauchy data arc functions of a Holmgren class 2! A function g( t ) defined on (α , β ) is said to be of a Holmgren class 2, if g ϵ C ∞ (α, β) and for all nonnegative integers n there exist positive constants c and s such that |g (n) | < cs n (2 n )!.