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Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms

Year 2022, , 191 - 212, 30.06.2022
https://doi.org/10.53006/rna.1082465

Abstract

In this paper, we study the Robin-Dirichlet
problem $(P_{n})$ for a strongly damped wave equation with
arithmetic-mean terms $S_{n}u$ and $\hat{S}_{n}u,$ where
$u$ is the unknown function, $S_{n}u=\tfrac{1}{n}
\sum\nolimits_{i=1}^{n}u(\tfrac{i-1}{n},t)$ and $\hat{S}_{n}u=
\tfrac{1}{n}\sum\nolimits_{i=1}^{n}u_{x}^{2}(\tfrac{i-1}{n},t)$.
First, under suitable conditions, we prove that, for each $n\in
\mathbb{N},$ $(P_{n})$ has a unique weak solution $u^{n}$. Next, we prove that the sequence of solutions $u^{n}$ converge strongly in appropriate spaces to the weak solution $u$ of the problem $(P),$ where $(P)$ is defined by $(P_{n})$ in which the arithmetic-mean terms $S_{n}u$ and $\hat{S}
_{n}u$ are replaced by $\int\nolimits_{0}^{1}u(y,t)dy$ and
$\int\nolimits_{0}^{1}u_{x}^{2}(y,t)dy,$ respectively. Finally,
some remarks on a couple of open problems are given.

References

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  • [2] M.M. Cavalcanti, V.N.D. Cavalcanti, J.S. Prates Filho, J.A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, J. Math. Anal. Appl. 226 (1) (1998) 40-60.
  • [3] M.M. Cavalcanti, V.N.D. Cavalcanti, J.A. Soriano, J.S. Prates Filho, Existence and asymptotic behaviour for a degenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions, Revista Matematica Complutense, 14 (1) (2001) 177-203.
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  • [6] G.R. Kirchhoff, Vorlesungen über Mathematische Physik: Mechanik, Teuber, Leipzig, 1876, Section 29.7.
  • [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.
  • [8] P. Massat, Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Eqs. 48 (1983) 334-349.
  • [9] N.H. Nhan, L.T.P. Ngoc, N.T. Long, Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term, Bound. Value Prob. (2017) 2017: 87.
  • [10] Vittorino Pata, Marco Squassina, On the strongly damped wave equation, Commun. Math. Phys. 253 (2005) 511-533.
  • [11] M. Pellicer, J. Solà-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl. 294 (2004) 687-698.
  • [12] M. Pellicer, J. Solà-Morales, Spectral analysis and limit behaviours in a spring-mass system, Comm. Pure. Appl. Math. 7 (3) (2008) 563-577.
  • [13] R.E. Showalter, Hilbert space methods for partial differential equations, Elec. J. Diff. Eqns. Monograph 01, 1994.
  • [14] G. Todorova, E. Vitillaro, Blow-up for nonlinear dissipative wave, J. Math. Anal. Appl. 303 (2005) 242-257.
Year 2022, , 191 - 212, 30.06.2022
https://doi.org/10.53006/rna.1082465

Abstract

References

  • [1] G.F. Carrier, On the nonlinear vibrations problem of elastic string, Quart. J. Appl. Math. 3 (2) (1945) 157-165.
  • [2] M.M. Cavalcanti, V.N.D. Cavalcanti, J.S. Prates Filho, J.A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, J. Math. Anal. Appl. 226 (1) (1998) 40-60.
  • [3] M.M. Cavalcanti, V.N.D. Cavalcanti, J.A. Soriano, J.S. Prates Filho, Existence and asymptotic behaviour for a degenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions, Revista Matematica Complutense, 14 (1) (2001) 177-203.
  • [4] M.M. Cavalcanti, V.N.D. Cavalcanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Diff. Eqns. 6 (6) (2001) 701-730.
  • [5] O.M. Jokhadze, Global Cauchy problem for wave equations with a nonlinear damping term, Differential Equations, 50 (1) (2014) 57-65.
  • [6] G.R. Kirchhoff, Vorlesungen über Mathematische Physik: Mechanik, Teuber, Leipzig, 1876, Section 29.7.
  • [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969.
  • [8] P. Massat, Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Eqs. 48 (1983) 334-349.
  • [9] N.H. Nhan, L.T.P. Ngoc, N.T. Long, Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term, Bound. Value Prob. (2017) 2017: 87.
  • [10] Vittorino Pata, Marco Squassina, On the strongly damped wave equation, Commun. Math. Phys. 253 (2005) 511-533.
  • [11] M. Pellicer, J. Solà-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl. 294 (2004) 687-698.
  • [12] M. Pellicer, J. Solà-Morales, Spectral analysis and limit behaviours in a spring-mass system, Comm. Pure. Appl. Math. 7 (3) (2008) 563-577.
  • [13] R.E. Showalter, Hilbert space methods for partial differential equations, Elec. J. Diff. Eqns. Monograph 01, 1994.
  • [14] G. Todorova, E. Vitillaro, Blow-up for nonlinear dissipative wave, J. Math. Anal. Appl. 303 (2005) 242-257.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Le Thi Phuong Ngoc

Nguyen Vu Dzung

Nguyen Huu Nhan

Nguyen Thanh Long

Publication Date June 30, 2022
Published in Issue Year 2022

Cite

APA Ngoc, L. T. P., Dzung, N. V., Nhan, N. H., Long, N. T. (2022). Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms. Results in Nonlinear Analysis, 5(2), 191-212. https://doi.org/10.53006/rna.1082465
AMA Ngoc LTP, Dzung NV, Nhan NH, Long NT. Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms. RNA. June 2022;5(2):191-212. doi:10.53006/rna.1082465
Chicago Ngoc, Le Thi Phuong, Nguyen Vu Dzung, Nguyen Huu Nhan, and Nguyen Thanh Long. “Existence, Uniqueness, and Convergence of Solutions of Strongly Damped Wave Equations With Arithmetic-Mean Terms”. Results in Nonlinear Analysis 5, no. 2 (June 2022): 191-212. https://doi.org/10.53006/rna.1082465.
EndNote Ngoc LTP, Dzung NV, Nhan NH, Long NT (June 1, 2022) Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms. Results in Nonlinear Analysis 5 2 191–212.
IEEE L. T. P. Ngoc, N. V. Dzung, N. H. Nhan, and N. T. Long, “Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms”, RNA, vol. 5, no. 2, pp. 191–212, 2022, doi: 10.53006/rna.1082465.
ISNAD Ngoc, Le Thi Phuong et al. “Existence, Uniqueness, and Convergence of Solutions of Strongly Damped Wave Equations With Arithmetic-Mean Terms”. Results in Nonlinear Analysis 5/2 (June 2022), 191-212. https://doi.org/10.53006/rna.1082465.
JAMA Ngoc LTP, Dzung NV, Nhan NH, Long NT. Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms. RNA. 2022;5:191–212.
MLA Ngoc, Le Thi Phuong et al. “Existence, Uniqueness, and Convergence of Solutions of Strongly Damped Wave Equations With Arithmetic-Mean Terms”. Results in Nonlinear Analysis, vol. 5, no. 2, 2022, pp. 191-12, doi:10.53006/rna.1082465.
Vancouver Ngoc LTP, Dzung NV, Nhan NH, Long NT. Existence, uniqueness, and convergence of solutions of strongly damped wave equations with arithmetic-mean terms. RNA. 2022;5(2):191-212.