ИСТИНА

The main goal of our work is studying of the asymptotic behavior of the solutions of Gurtin-Pipkin type integro-differential equations on the base of spectral analysis of their symbols. For this purpose, strong solutions of these equations are represented as a sum of terms corresponding to the real and nonreal parts of the spectrum of the operator functions that are the symbols of these equations. The resulting representations are new for the given class of integro-differential equations. Since the Gurtin-Pipkin type integro-differential equations arise in numerous applications, it is reasonable and natural to consider such equations with operator coefficients in a Hilbert space (abstract integro-differential equations), which, if necessary, can be specified as integro-differential equations with partial derivatives with respect to spatial variables.