## Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 9966-9989 |

Number of pages | 24 |

Journal | Applied Mathematical Modelling |

Volume | 40 |

Issue number | 23-24 |

DOIs | |

State | Published - 01 12 2016 |

### Bibliographical note

Publisher Copyright:© 2016 Elsevier Inc.

## Keywords

- Analytic semigroup
- Global solvability
- Inverse function theorem
- Sectorial operator
- Volterra integro-differential equation