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 φ 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, φ) 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.