J. Comp. Math., 1 (1983), pp. 148-160. Estimation For Solutions Of Ill-Posed Cauchy Problems Of Differential Equations With Pseudo-Differential Operators Guan-Quan Zhang 11 Computing Center, Academia Sinica, China Received 1982-9-30 Revised Online 2006-11-17 Abstract In this paper we discuss the estimation for solutions of the ill-possed 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.(pseduo-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 funtions in the Three-Circle thoeorem of Hadamard. Another is the estimate of the growth rate of ||u(t)|| when $A(1)u(1)\in H$. Key words: