Global existence for a semi-linear Volterra parabolic equation and neutral system with infinite delay

This paper studies the global and local existence of classical solutions for a semilinear Volterra integro-differential equation of parabolic type: (u + k * u)' = A(u k * u) + f(u) + g, where A is a (not necessarily densely defined) sectorial operator with its spectrum contained in the left half plane. We transform the Volterra equation into a neutral system with infinite delay assuming the history phi of the system is known. The inverse function theorem is then employed to prove the global existence of classical solution to the system for appropriate "small" data (g, phi) if 0 belongs to the resolvent set of A. An example of the linear part being non-densely defined elliptic operators is shown to illustrate the existence theorems, and an application of our results to compressible viscoelastic fluids with hereditary viscosity is also addressed. (C) 2016 Elsevier Inc. All rights reserved.