A Noncharacteristic Cauchy Problem for Linear Parabolic Equations I: Solvability

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 )!.