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ESTIMATION FOR SOLUTIONS OF ILL-POSED CAUCHY PROBLEMS OF DIFFERENTIAL EQUATIONS WITH PSEUDO-DIFFERENTIAL OPERATORS: Part I. Case of First Order Operators

Zhang Guan-quan and 张关泉
Journal of Computational Mathematics
Vol. 1, No. 2 (April 1983), pp. 148-160
Stable URL: http://www.jstor.org/stable/43692313
Page Count: 13
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
ESTIMATION FOR SOLUTIONS OF ILL-POSED CAUCHY PROBLEMS OF DIFFERENTIAL EQUATIONS WITH PSEUDO-DIFFERENTIAL OPERATORS: Part I. Case of First Order Operators
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Abstract

In this paper we discuss the estimation for solutions of the ill-posed Cauchy problems of the following differential equation $\frac{{du\left( t \right)}}{{dt}} = A\left( t \right)u\left( t \right) + N\left( t \right)u\left( t \right),\forall t \in \left( {0,1} \right)$ 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 ǁu(t)ǁ when A(1)u(1) ∊ H.

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