Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms

Radu, Petronela

doi:10.4064/am35-3-7

Cited by 11 publications

(19 citation statements)

References 22 publications

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“…32. Finally, pass to the limit in the weak variational form for Galerkin approximations to conclude the existence of a local weak solution to the original problem.…”

Section: Strategies and Technical Difficultiesmentioning

confidence: 99%

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Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

Pei

Rammaha

Toundykov

2015

Journal of Mathematical Physics

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Articles you may be interested inBlow up of a solution for a system of nonlinear higher-orderwave equations with strong damping terms AIP Conf.Discrete rotating waves in a ring of coupled mechanical oscillators with strong damping Uniqueness of solutions to the helically reduced wave equation with Sommerfeld boundary conditionsThis paper investigates a quasilinear wave equation with Kelvin-Voigt damping,, in a bounded domain Ω ⊂ R 3 and subject to Dirichlét boundary conditions. The operator ∆ p , 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f (u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W 1, p 0 (Ω) into L 2 (Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy. C 2015 AIP Publishing LLC.

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“…32. Finally, pass to the limit in the weak variational form for Galerkin approximations to conclude the existence of a local weak solution to the original problem.…”

Section: Strategies and Technical Difficultiesmentioning

confidence: 99%

“…Consider a sequence of smooth cut-off functions η n , as introduced in Ref. 32. More precisely, we choose a sequence η n ∈ C ∞ 0 (R) that satisfies…”

Section: Local Solution For More General Sourcesmentioning

confidence: 99%

Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

Pei

Rammaha

Toundykov

2015

Journal of Mathematical Physics

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“…In order to establish the existence of solutions for more general sources we employ another truncation argument as in [26,24]. To begin, select as in [25] With these truncated sources we intend to build a sequence {u n } of approximate solutions where each u n satisfies the corresponding n-problem…”

Section: General Sourcesmentioning

confidence: 99%

On wave equations of the p-Laplacian type with supercritical nonlinearities

Kass

Rammaha

2019

Nonlinear Analysis

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This article focuses on a quasilinear wave equation of p-Laplacian type:in a bounded domain Ω ⊂ R 3 with a sufficiently smooth boundary Γ = ∂Ω subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator ∆ p , 2 < p < 3, denotes the classical p-Laplacian. The interior and boundary terms f (u), h(u) are sources that are allowed to have a supercritical exponent, in the sense that their associated Nemytskii operators are not locally Lipschitz from W 1,p (Ω) into L 2 (Ω) or L 2 (Γ). Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time, provided the damping terms dominates the corresponding sources in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

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“…is similar; see [28], [3], [23], [4], [24], [5], [6]. The exponents p and m in equation (1.1) need to satisfy p + p m < 6.…”

Section: Introductionmentioning

confidence: 99%

On the regularizing effect of nonlinear damping in hyperbolic equations

Todorova

Yordanov

2015

Trans. Amer. Math. Soc.

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Global well-posedness in H 2 (R 3 ) × H 1 (R 3 ) is shown for nonlinear wave equations of the form u + f (u) + g(u t ) = 0, where t ∈ R + . The main assumption is that the nonlinear damping g(u t ) behaves like |u t | m−1 u t with m ≥ 2 and the defocusing nonlinearity f (u) is like |u| p−1 u with p ≥ 2. The result also applies to certain exponential functions, such as f (u) = sinh u. It is observed that the nonlinear damping gives rise to a new monotone quantity involving the second-order derivatives of u and leading to a priori estimates for initial data of any size.Global well-posedness in H 1 (R 3 ) × L 2 (R 3 ) is shown for the same equation in the critical case f (u) = u 5 and g(u t ) = |u t | 2/3 u t . The main tool is a new estimate for the solution of the nonlinear equation in L 4 (R + , L 12 (R 3 )).

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Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms

Cited by 11 publications

References 22 publications

Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

On wave equations of the p-Laplacian type with supercritical nonlinearities

On the regularizing effect of nonlinear damping in hyperbolic equations

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