Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions

TL;DR

This paper investigates a viscoelastic wave equation with dynamic boundary conditions, establishing conditions for global existence, exponential growth, or finite-time blow-up of solutions.

Contribution

It introduces a combined analytical approach to prove existence, uniqueness, and long-term behavior of solutions for a complex wave model with boundary dynamics.

Findings

Global existence under certain initial data restrictions

Exponential growth when interior source dominates boundary damping

Finite-time blow-up without strong damping

Abstract

The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.