On a nonlinear problem with Dirichlet and Acoustic boundary conditions

Alcântara, Adriano A.; Carmo, Bruno A.; Clark, H. R.; Guardia, R. R.; Rincon, Mauro A.

doi:10.1016/j.amc.2021.126514

Cited by 5 publications

(10 citation statements)

References 37 publications

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“…Additionally, the solution blows up roughly when the interior source dominates the interior damping term and if the boundary source dominates the boundary damping. For related results, including numerical implementation and Laplace-Beltrami operator instead of ∆, we refer the reader to [1]- [2], [9]- [10], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

Bautista

Límaco

Potenciano-Machado

2023

Preprint

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In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained by Frota-Goldstein [18] and Limaco-Clark-Frota-Medeiros [26] to allow weak internal dampings and removing the dimensional restriction 1≤ n≤4. The method we use is based on a finite-dimensional approach by combining the Faedo-Galerkin method with suitable energy estimates and multiplier techniques.

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Section: Introductionmentioning

confidence: 99%

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

Bautista

Límaco

Potenciano-Machado

2023

Preprint

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“…Remarkable examples of nonlinearities with weak and strong dissipative mechanisms were considered, including damping, delay, and memory. In this direction, we quote the results in [5,[7][8][9][15][16][17][18][19][20][21]21,22,22,23,[23][24][25][26][27][28][29][30][31][32][33][34] and the references therein. The main idea in proving the works mentioned above is getting a priori estimates in appropriate finite-dimensional Hilbert spaces to perform a Faedo-Galerkin method combined with multipliers methods (energy estimates) and semigroup theory.…”

Section: Introductionmentioning

confidence: 99%

\Delta

, we refer the reader to [8–11] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

Bautista,

Límaco,

Potenciano‐Machado

2023

Math Methods in App Sciences

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In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction . The method we use is based on a finite‐dimensional approach by combining the Faedo‐Galerkin method with suitable energy estimates and multiplier techniques.

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“…Após estes trabalhos, o primeiro não linear foi devido a Frota e Goldstein [10] em 2000. A partir deste surgiram muitos outros importantes trabalhos (com versão linear e não linear no domínio ou fronteira) envolvendo condições de fronteira Acústica cujas as referências podem ser vista em Adriano et al [4].…”

Section: Introductionunclassified

“…A tese [1] é o primeiro trabalho a contemplar a análise e simulação numérica para equações de ondas não linear com coeficientes variáveis submetidas a condições de fronteira Acústica com não linearidades gerais no domínio e na fronteira. Destacamos que, para os três modelos em [1], publicamos em [2], [4] e [3] os resultados relacionados a existência e unicidade de solução e o decaimento assintótico da energia.…”

Section: Introductionunclassified

Simulação numérica para um modelo de ondas com condições de fronteira do tipo Dirichlet, Acústica e Impenetrabilidade

A. Alcântara,

A. Rincon,

R. Clark

2023

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Este artigo consiste de uma pequena versão da tese de doutorado relacionada a análise e simulação numérica para três modelos de ondas com condições de fronteira do tipo Dirichlet, Acústica e Impenetrabilidade. Apresentamos aqui o modelo mais geral dentre os três com os resultados de análise e simulação numérica. Para as simulações numéricas, usamos o método dos elementos finitos no espaço (com base linear e quadrática de Lagrange), o método de Crank-Nicolson no tempo e, para cada tempo discreto, o método de Newton é usado para resolver o sistema algébrico não linear. Além disso, a ordem de convergência (sub-ótimo e ótimo) são apresentadas numericamente.

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On a nonlinear problem with Dirichlet and Acoustic boundary conditions

Cited by 5 publications

References 37 publications

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

Well‐posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

Simulação numérica para um modelo de ondas com condições de fronteira do tipo Dirichlet, Acústica e Impenetrabilidade

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