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{\bf COEFFICIENT STABILITY OF THE SOLUTION FOR THE CAUCHY PROBLEM
FOR DIFFERENTIAL--OPERATOR AND OPERATOR--DIFFERENCE EQUATIONS}
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{\bf A.A.~Samarskii and P.N.~Vabishchevich}

{\it Institute for Mathematical Modelling,
Russian Academy of Sciences}
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{\bf P.P.~Matus}

{\it Institute of Mathematics
Belarus National Academy of Sciences}
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In studying initial/boundary value problems for time-dependent
equations of mathematical physics attention is focused on
the solution stability with respect to the initial data and
the right hand side (RHS). In a more general situation it is necessary
to require continuous dependence of the solution on a perturbation
of problem operators (e.g. coefficients), too. In this case
there are told about the strong stability.

Apriori estimates which demonstrate continuous dependence of the
problem solution on perturbations of the RHS and the operator
were obtained at various conditions for steady-state problems
(operator equations of the first kind). Transient problems
are less well investigated.

This work deals with derivation of stability estimates for perturbations
of the operator, RHS and the initial condition of the Cauchy problem for
evolutionary equations considered in Hilbert spaces. Apriori estimates
are obtained for the error under the natural assumptions on perturbation
of the problem operator. Conducting discretization in time
we obtain the operator-difference equation. The simplest apriori estimates
of the strong stability consistent with the corresponding estimate
for the differential-operator equation are presented. The basic results
are illustrated on the sample of the initial/boundary value
problem for the one-dimensional parabolic equation.

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