Abstract
The paper studies the global existence and general decay of solutions using Lyapunov functional for a nonlinear wave equation, taking into account the fractional derivative boundary condition and memory term. In addition, we establish the blow-up of solutions with nonpositive initial energy.
Highlights
Extraordinary differential equations, known as fractional differential equations, are a generalization of differential equations through fractional calculus
Much attention has been accorded to fractional partial differential equations during the past two decades due to the many chemical engineering, biological, ecological, and electromagnetism phenomena that are modeled by initial boundary value problems with fractional boundary conditions
We consider three different cases on the sign of the initial energy as recently examined by Zarai et al [17], where they studied the blow-up of a system of nonlocal singular viscoelastic equations
Summary
Extraordinary differential equations, known as fractional differential equations, are a generalization of differential equations through fractional calculus. Note that the nonlinear wave equation with boundary fractional damping case was first considered by authors in [4], where they used the augmented system to prove the exponential stability and blow-up of solutions in finite time. Motivated by our recent work in [4] and based on the construction of a Lyapunov function, we prove in this paper under suitable conditions on the initial data the stability of a wave equation with fractional damping and memory term. 1406, Lemma 4.2) Let J(t) be a nonincreasing function on [t0, ∞) verifying the differential inequality
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