Nonlinear Volterra equations for heat flow in materials with memory

The nonlinear Volterra equation u(t)+(b*Au)(t) not an element of f(t) is considered. Existing and recent results are discussed for the following problems: (1) the global existence and uniqueness of solutions and their continuous dependence on the data; (2) the boundedness and asymptotic behavior as it approaches infinity in the special cases where X = H is a real Hilbert space and A is either a maximum monotone operator on H, or A is a subdifferential of a proper, convex, lower semicontinuum function; and (3) the existence, boundedness, and asymptotic behavior of positive solutions in the general setting. The theory is used to study one possible model problem for heat flow in a material with memory which can be transformed to the equivalent form under physically reasonable assumptions. This and other models for heat flow in such materials are formulated from physical principals and discussed.