General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

Lv, Mengxian; Hao, Jianghao

doi:10.3934/mcrf.2021058

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…Moreover, they proved the finite time blow-up result of solutions with negative initial energy. Recently, the following coupled viscoelastic wave equations of Kirchhoff-type with dynamic boundary conditions have been studied by Lv and Hao [24]:…”

Section: Constant Exponentsmentioning

confidence: 99%

“…Moreover, they proved the finite time blow‐up result of solutions with negative initial energy. Recently, the following coupled viscoelastic wave equations of Kirchhoff‐type with dynamic boundary conditions have been studied by Lv and Hao [24]:

({, \begin{matrix} right left \\ array & array {(u_{i})}_{t t} - Δ {(u_{i})}_{t} - M (\sum_{i = 1}^{l} ‖ \nabla u_{i} ‖^{2}) Δ u_{i} + \int_{0}^{t} g_{1} (t - s) Δ u_{i} (s) d s + η (t) h ({(u_{i})}_{t}) = 0, in Ω \times (0, \infty) \\ array & array u_{i} (x, t) = 0, on Γ_{0} \times (0, T) \\ array & array {(u_{i})}_{t t} + \partial_{ν} {(u_{i})}_{t} + M (\sum_{i = 1}^{l} ‖ \nabla u_{i} ‖^{2}) \partial_{ν} u_{i} - \int_{0}^{t} g_{1} (t - s) \partial_{ν} u_{i} (s... \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

Shahrouzi,

Ferreira,

Tahamtani

2023

Math Methods in App Sciences

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This work studies the global behavior of solutions for a system of ‐Kirchhoff‐type equations. We prove the global existence of solutions, and next, by constructing suitable auxiliary functionals, the general decay of solutions has been confirmed when the exponents satisfy appropriate conditions. Also, under suitable conditions on data, we establish the finite time blow‐up of solutions with negative initial energy. Our results extend and improve many earlier results in the literature.

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Section: Constant Exponentsmentioning

confidence: 99%

({, \begin{matrix} right left \\ array & array {(u_{i})}_{t t} - Δ {(u_{i})}_{t} - M (\sum_{i = 1}^{l} ‖ \nabla u_{i} ‖^{2}) Δ u_{i} + \int_{0}^{t} g_{1} (t - s) Δ u_{i} (s) d s + η (t) h ({(u_{i})}_{t}) = 0, in Ω \times (0, \infty) \\ array & array u_{i} (x, t) = 0, on Γ_{0} \times (0, T) \\ array & array {(u_{i})}_{t t} + \partial_{ν} {(u_{i})}_{t} + M (\sum_{i = 1}^{l} ‖ \nabla u_{i} ‖^{2}) \partial_{ν} u_{i} - \int_{0}^{t} g_{1} (t - s) \partial_{ν} u_{i} (s... \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

Shahrouzi,

Ferreira,

Tahamtani

2023

Math Methods in App Sciences

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show abstract

“…He [16] reported some uniform decaying results for the energy associated to IBVP (4) under some other conditions. The other more interesting existence and stability results concerning viscoelastic (quasi-)linear wave equations could be seen in References [17][18][19][20][21][22][23][24][25][26][27][28][29] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

Wang

Zhao³

et al. 2023

Axioms

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Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper.

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General decay results for a viscoelastic wave equation with the logarithmic nonlinear source and dynamic Wentzell boundary condition

Guo,

Zhang

2024

Nonlinear Analysis: Real World Applications

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General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

Cited by 4 publications

References 21 publications

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

General decay results for a viscoelastic wave equation with the logarithmic nonlinear source and dynamic Wentzell boundary condition

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