Solutions to Second-Order Nonlocal Evolution Equations Governed by Non-Autonomous Forms

TL;DR

This paper establishes conditions for solutions to second-order nonlocal evolution equations with non-autonomous forms, applying fixed-point methods and illustrating with PDEs like viscoelastic membranes.

Contribution

It provides new sufficient conditions for existence of solutions to complex nonlocal PDEs with time-dependent properties using advanced analytical techniques.

Findings

Proved existence conditions for second-order nonlocal evolution equations.

Applied theoretical results to PDEs modeling viscoelastic membranes.

Demonstrated the effectiveness of fixed-point methods in non-autonomous settings.

Abstract

Our main contributions include proving sufficient conditions for the existence of solution to a second order problem with nonzero nonlocal initial conditions, and providing a comprehensive analysis using fundamental solutions and fixed-point techniques. The theoretical results are illustrated through applications to partial differential equations, including vibrating viscoelastic membranes with time-dependent material properties and nonlocal memory effects.