General decay of solutions of a nonlinear system of viscoelastic wave equations

Said‐Houari, Belkacem; Messaoudi, Salim A.; Guesmia, Aı̈ssa

doi:10.1007/s00030-011-0112-7

Cited by 56 publications

(37 citation statements)

References 27 publications

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“…Lemma Suppose that holds. Then, there exist η > 0 such that for any

u, v \in H^{2} false(normalΩ false) \cap H_{0}^{1} false(normalΩ false),

we have

{(‖‖, u + v)}_{2 false(r + 2 false)}^{2 false(r + 2 false)} + 2 {(‖‖, u v)}_{r + 2}^{r + 2} \leq η {false(l_{1} {(‖‖, \nabla u)}_{2}^{2} + l_{2} {(‖‖, \nabla v)}_{2}^{2} false)}^{r + 2},

is satisfied.…”

Section: Preliminariesmentioning

confidence: 99%

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

Pışkın

Ekinci

2019

Math Methods in App Sciences

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In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions gi(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.

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“…Lemma Suppose that holds. Then, there exist η > 0 such that for any

u, v \in H^{2} false(normalΩ false) \cap H_{0}^{1} false(normalΩ false),

we have

{(‖‖, u + v)}_{2 false(r + 2 false)}^{2 false(r + 2 false)} + 2 {(‖‖, u v)}_{r + 2}^{r + 2} \leq η {false(l_{1} {(‖‖, \nabla u)}_{2}^{2} + l_{2} {(‖‖, \nabla v)}_{2}^{2} false)}^{r + 2},

is satisfied.…”

Section: Preliminariesmentioning

confidence: 99%

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

Pışkın

Ekinci

2019

Math Methods in App Sciences

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“…We suppose that Then, we can get Remark 1. 27 There exist two constants c 0 > 0 and c 1 > 0 such that…”

Section: Assumptions and Local Existencementioning

confidence: 99%

Existence and decay rates for a coupled Balakrishnan‐Taylor viscoelastic system with dynamic boundary conditions

Feng

Soufyane

2020

Math Methods in App Sciences

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In this paper, we are concerned with a coupled viscoelastic wave system with Balakrishnan‐Taylor dampings, dynamic boundary conditions, source terms, and past histories. Under suitable assumptions on relaxation functions and source terms, we prove the global existence of solutions by potential well theory and we establish a more general decay result of energy, in which the exponential decay and polynomial decay are only special cases, by introducing suitable energy and perturbed Lyapunov functionals.

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“…Furthermore, they proved that the weak solutions blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms. Moreover, many studies of the global well-posedness for wave systems with dissipative terms have been researched in [24][25][26][27][28].…”

Section: Historical Researchmentioning

confidence: 99%

“…By reviewing above known results and also [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], we will face the fact that the following unsolved problems arise naturally. Firstly, from [38] we know the global existence for the definitely positive energy, but we know less for the initial energy which may be negative.…”

Section: Unsolved Problemsmentioning

confidence: 99%

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

Jiang

2017

Advances in Mathematical Physics

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In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars' attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.

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General decay of solutions of a nonlinear system of viscoelastic wave equations

Cited by 56 publications

References 27 publications

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

Existence and decay rates for a coupled Balakrishnan‐Taylor viscoelastic system with dynamic boundary conditions

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

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