Estimation for Solutions of Ill-Posed Cauchy Problems of Differential Equations with Pseudo-Differential Operators
Year: 1983
Journal of Computational Mathematics, Vol. 1 (1983), Iss. 2 : pp. 148–160
Abstract
In this paper we discuss the estimation for solutions of the ill-posed Cauchy problems of the following differential equation$\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$, where A(t) is a p. d. o. (pseudo-differential operator(s)) of order 1 or 2, N(t) is a uniformly bounded $H-›H$ linear operator. It is proved that if the symbol of the principal part of A(t) satisfies certain algebraic conditions, two estimates for the solution u(t) hold. One is similar to the estimate for analytic functions in the Three-Circle Theorem of Hadamard. Another is the estimate of the growth rate of ||u(t)|| when $A(1)u(1)\in H$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/1983-JCM-9691
Journal of Computational Mathematics, Vol. 1 (1983), Iss. 2 : pp. 148–160
Published online: 1983-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13