June  2020, 9(2): 487-508. doi: 10.3934/eect.2020021

On a Kirchhoff wave model with nonlocal nonlinear damping

Nucleus of Exact and Technological Sciences, State University of Mato Grosso do Sul, 79804-970 Dourados, MS, Brazil

Received  October 2018 Revised  June 2019 Published  December 2019

Fund Project: Partially supported by FUNDECT Grant 219/2016

This paper is concerned with the well-posedness as well as the asymptotic behavior of solutions for a quasi-linear Kirchhoff wave model with nonlocal nonlinear damping term $ \sigma\left(\int_{\Omega}|\nabla u|^2\,dx\right )g(u_t), $ where $ \sigma $ and $ g $ are nonlinear functions under proper conditions. The analysis of such a damping term is presented for this kind of Kirchhoff models and consists the main novelty in the present work.

Citation: Vando Narciso. On a Kirchhoff wave model with nonlocal nonlinear damping. Evolution Equations & Control Theory, 2020, 9 (2) : 487-508. doi: 10.3934/eect.2020021
References:
[1]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

[2]

M. M. CavalcantiV. N. Domingos CavalcantiJ. A. Soriano and J. S. Prates Filho, Existence ans asymptotic behaviour for a dgenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions, Rev. Mat. Complut., 14 (2001), 177-203.  doi: 10.5209/rev_REMA.2001.v14.n1.17054.  Google Scholar

[3]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.   Google Scholar

[4]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar

[5]

A. T. CousinC. L. Frota and N. A. Larkin, Global solvability and decay of the energy for the nonhomogeneous Kirchhoff equation, Differential and Integral Equations, 15 (2002), 1219-1236.   Google Scholar

[6]

M. Ghisi, Global solutions for dissipative Kirchhoff string with non-Lipschitz nonlinear term, J. Differential Equations, 230 (2006), 128-139.  doi: 10.1016/j.jde.2006.07.020.  Google Scholar

[7]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.  Google Scholar

[8]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155.  doi: 10.1016/j.jde.2009.04.010.  Google Scholar

[9]

G. Kirchhoff, Vorlessungen über Mechanik, Tauber Leipzig, 1883. Google Scholar

[10]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.   Google Scholar

[11]

I. Lasiecka and J. Ong, Global solvability and uniform decays of solution to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107.  doi: 10.1080/03605309908821495.  Google Scholar

[12]

P. Lazo, Global solution for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.  doi: 10.1016/j.amc.2007.11.056.  Google Scholar

[13]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, (French) Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[14]

J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18 Dunod, Paris, 1968.  Google Scholar

[15]

T. Matsuyama and R. Ikehara, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.  Google Scholar

[16]

L. A. Medeiros and M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231.   Google Scholar

[17]

G. P. Menzala and J. M. Pereira, On smooth global solutions of a Kirchhoff type equation on unbounded domains, Differential Integral Equations, 8 (1995), 1571-1583.   Google Scholar

[18]

M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.  doi: 10.2206/kyushumfs.30.257.  Google Scholar

[19]

M. Nakao, On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z., 193 (1986), 227-234.  doi: 10.1007/BF01174332.  Google Scholar

[20]

M. Nakao, Global attractors for wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.  doi: 10.1016/j.jde.2005.09.013.  Google Scholar

[21]

M. Nakao, Local attractors for nonlinear wave equations with some dissipative terms, Far East J. Math. Sci. (FJMS), 27 (2007), 497-515.   Google Scholar

[22]

M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009), 652-659.  doi: 10.1016/j.jmaa.2008.09.010.  Google Scholar

[23]

M. Nakao and Z. J. Yang, Global attractor for some quasi-linear wave equation with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.   Google Scholar

[24]

K. Ono, Global existence, decay, and blow up of solution for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301.  doi: 10.1006/jdeq.1997.3263.  Google Scholar

[25]

T. Taniguchi, Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms, J. Math. Anal. Appl., 361 (2010), 566-578.  doi: 10.1016/j.jmaa.2009.07.010.  Google Scholar

[26]

Z. J. Yang, Long-time behavior of the Kirchhoff type equation with strong damping in $\mathbb{R}^N$, J. Differential Equations, 242 (2007), 269-286.  doi: 10.1016/j.jde.2007.08.004.  Google Scholar

[27]

Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation,, J. Differential Equations, 249 (2010) 3258–3278. doi: 10.1016/j.jde.2010.09.024.  Google Scholar

[28]

Z. J. Yang and Y. N. Li, Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653.  doi: 10.3934/dcds.2018111.  Google Scholar

[29]

Z. J. Yang and W. Yunqing, Stability of attractor for the Kirchhoff wave equation with strong damping and critical nonlinearities, J. Math. Anal. Appl., 469 (2019), 298-320.  doi: 10.1016/j.jmaa.2018.09.012.  Google Scholar

show all references

References:
[1]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

[2]

M. M. CavalcantiV. N. Domingos CavalcantiJ. A. Soriano and J. S. Prates Filho, Existence ans asymptotic behaviour for a dgenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions, Rev. Mat. Complut., 14 (2001), 177-203.  doi: 10.5209/rev_REMA.2001.v14.n1.17054.  Google Scholar

[3]

I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106.   Google Scholar

[4]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar

[5]

A. T. CousinC. L. Frota and N. A. Larkin, Global solvability and decay of the energy for the nonhomogeneous Kirchhoff equation, Differential and Integral Equations, 15 (2002), 1219-1236.   Google Scholar

[6]

M. Ghisi, Global solutions for dissipative Kirchhoff string with non-Lipschitz nonlinear term, J. Differential Equations, 230 (2006), 128-139.  doi: 10.1016/j.jde.2006.07.020.  Google Scholar

[7]

M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.  Google Scholar

[8]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155.  doi: 10.1016/j.jde.2009.04.010.  Google Scholar

[9]

G. Kirchhoff, Vorlessungen über Mechanik, Tauber Leipzig, 1883. Google Scholar

[10]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533.   Google Scholar

[11]

I. Lasiecka and J. Ong, Global solvability and uniform decays of solution to quasilinear equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 24 (1999), 2069-2107.  doi: 10.1080/03605309908821495.  Google Scholar

[12]

P. Lazo, Global solution for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601.  doi: 10.1016/j.amc.2007.11.056.  Google Scholar

[13]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, (French) Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[14]

J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18 Dunod, Paris, 1968.  Google Scholar

[15]

T. Matsuyama and R. Ikehara, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.  Google Scholar

[16]

L. A. Medeiros and M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complut. Madrid, 3 (1990), 213-231.   Google Scholar

[17]

G. P. Menzala and J. M. Pereira, On smooth global solutions of a Kirchhoff type equation on unbounded domains, Differential Integral Equations, 8 (1995), 1571-1583.   Google Scholar

[18]

M. Nakao, Convergence of solutions of the wave equation with a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (1976), 257-265.  doi: 10.2206/kyushumfs.30.257.  Google Scholar

[19]

M. Nakao, On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions, Math. Z., 193 (1986), 227-234.  doi: 10.1007/BF01174332.  Google Scholar

[20]

M. Nakao, Global attractors for wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229.  doi: 10.1016/j.jde.2005.09.013.  Google Scholar

[21]

M. Nakao, Local attractors for nonlinear wave equations with some dissipative terms, Far East J. Math. Sci. (FJMS), 27 (2007), 497-515.   Google Scholar

[22]

M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009), 652-659.  doi: 10.1016/j.jmaa.2008.09.010.  Google Scholar

[23]

M. Nakao and Z. J. Yang, Global attractor for some quasi-linear wave equation with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.   Google Scholar

[24]

K. Ono, Global existence, decay, and blow up of solution for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301.  doi: 10.1006/jdeq.1997.3263.  Google Scholar

[25]

T. Taniguchi, Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms, J. Math. Anal. Appl., 361 (2010), 566-578.  doi: 10.1016/j.jmaa.2009.07.010.  Google Scholar

[26]

Z. J. Yang, Long-time behavior of the Kirchhoff type equation with strong damping in $\mathbb{R}^N$, J. Differential Equations, 242 (2007), 269-286.  doi: 10.1016/j.jde.2007.08.004.  Google Scholar

[27]

Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation,, J. Differential Equations, 249 (2010) 3258–3278. doi: 10.1016/j.jde.2010.09.024.  Google Scholar

[28]

Z. J. Yang and Y. N. Li, Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 38 (2018), 2629-2653.  doi: 10.3934/dcds.2018111.  Google Scholar

[29]

Z. J. Yang and W. Yunqing, Stability of attractor for the Kirchhoff wave equation with strong damping and critical nonlinearities, J. Math. Anal. Appl., 469 (2019), 298-320.  doi: 10.1016/j.jmaa.2018.09.012.  Google Scholar

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