Asympotic Properties of Solutions of Nonlinear Abstract Volterra Equations

摘要

The purpose of this paper is to develop a general theory which gives sufficient conditions in terms of the kernel b, the operator A, and the forcing term f for the solution u of (V) to be bounded on t greater than or = 0 but less than infinity and which further assures that the solution u tends to a limit u sub infinity as t approaches infinity; under certain conditions u sub infinity = 0, under others u sub infinity is the unique solution of an appropriate 'limit equation' associated with (V). As one special case of this theory we give a complete analysis of the boundedness and asymptotic properties of the solution of the above heat flow problem, under physically reasonable assumptions concerning the relaxation functions, the nonlinear operator, the initial temperature distribution, and the external heat supply.