TY - JOUR
T1 - Estimation for Solutions of  Ill-Posed Cauchy Problems of Differential Equations with Pseudo-Differential Operators
AU - Guan-Quan Zhang
JO - Journal of Computational Mathematics
VL - 2
SP - 148
EP - 160
PY - 1983
DA - 1983/01
SN - 1
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9691.html
KW - 
AB - <p style="text-align: justify;">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 $A(1)u(1)\in H$.</p>