General decay and blow up of solution for a nonlinear wave equation with a fractional boundary damping

Abstract

The paper deals with the study of global existence of solutions and the general decay in a bounded domain for nonlinear wave equation with fractional derivative boundary condition by using the Lyaponov functional. Furthermore, the blow up of solutions with nonpositive initial energy combined with a positive initial energy is established.

General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms

“…We prove in this paper under suitable conditions on the initial data the stability of a wave equation with fractional damping and memory term. This technique of proof was recently used by [4] to study the exponential decay of a system of nonlocal singular viscoelastic equations. Here we also considered 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.…”

“…In the context of boundary dissipations of fractional order problems, the main research focus is on asymptotic stability of solutions starting by writing the equations as an augmented system. Then, various techniques are used such as LaSalle's invariance principle and the multiplier method mixed with frequency domain (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], and [18]).…”

“…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. This technique of proof was recently used by [9] and [4] to study the exponential decay of a system of nonlocal singular viscoelastic equations.…”

On Existence of Sequences of Weak Solutions of Fractional Systems with Lipschitz Nonlinearity

“…Various natural phenomena are modeled mathematically through the FDEs, and this is evident in numerous areas of physics, engineering, chemistry, and other fields. The fractionalorder partial differential equations have several applications in many fields such as engineering, biophysics, physics, mechanics, chemistry, and biology (see [1][2][3][4][5][6][7]). More and more efforts have been made in the fractional calculus field especially in FDEs (see, for instance, [2,5,[8][9][10][11][12][13][14][27][28][29][30][31][32][33][34][35][36][37][38][39]).…”