A Nonlinear Functional Differential Equation in Banach Space with Applications to Materials with Fading Memory

摘要

The report studies a nonlinear functional differential equation in Banach space. This equation is an abstract form of the equations of motion for nonlinear materials with fading memory. Its basic structure is hyperbolic in character so that global smooth solutions should not be expected in general. Memory effects, however, may induce a dissipative mechanism which, although very subtle, is effective so long as the solution is small. The report shows that if the memory is dissipative in an appropriate sense, then the history value problem associated with our equation has a unique global smooth solution provided the initial history and forcing are suitably smooth and small. The proof combines a fixed point argument to establish local existence with a chain of global a priori 'energy-type' estimates. The abstract results are then applied to establish global existence of smooth solutions to certain history value problems associated with the motion of nonlinear materials with fading memory, under assumptions which are realistic within the framework of continuum mechanics.