Abstract
In this paper, we are considering the one-dimensional porous-elastic system with nonlinear localized damping acting in a arbitrary small region of the interval under consideration. Assuming appropriate assumptions on the nonlinear terms, we establish the energy decay of the corresponding system. To do this, we used the observability inequality obtained for the conservative system combined with a unique continuation property recently introduced in Ma et al. (Attractors for locally damped Bresse systems and a unique continuation property, 2021. arXiv:2102.12025) and the reduction principle (see Daloutli et al. in Discrete Contin Dyn Syst 2(1):67–94, 2009) where the problem of decay rates with nonlinear damping is reduced to an appropriate stabilizability inequality for the linear equation. This study generalizes and improves previous literature outcomes.
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References
Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 6, 507–533 (1993)
Ho, L.F.: Exact controllability of the one dimensional wave equation with locally distributed control. SIAM J. Control Optim. 28(3), 733–748 (1990)
Chen, G., Fulling, S.A., Narcowich, F.J., Sun, S.: Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51(1), 266–301 (1991)
Martinez, P.: Decay of solutions of the wave equation with a local highly degenerate dissipation. Asymptot. Anal. 19(1), 1–17 (1999)
Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15(2), 205–235 (1990)
Zuazua, E.: Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 70(4), 513–529 (1991)
Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control optim. 30(5), 1024–1065 (1992)
Santos, M.L., Almeida Júnior, D.S., Rodrigues, J.H., Falcão Nascimento, F.A.: Decay rates for Timoshenko system with nonlinear arbitrary localized damping. Differ. Integral Equ. 27, 1-2-1–26 (2014)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Falcão Nascimento, F.A., Lasiecka, I., Rodrigues, J.H.: Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping. Z. Angew. Math. Phys. 65(6), 1189–1206 (2013)
Daloutli, M., Lasiecka, I., Toundykov, D.: Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete Contin. Dyn. Syst. 2(1), 67–94 (2009)
Ma, T.F, Monteiro, R.M., Seminario-Huertas, P.N.: Attractors for locally damped Bresse systems and a unique continuation property. arXiv:2102.12025 (2021)
Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16, 487–491 (2003)
Magaña, A., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials. Int. J. Solids Struct. 43, 3414–3427 (2006)
Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman and Hall/CRC, Boca Raton (1999)
Muñoz Rivera, J.E., Quintanilla, R.: On the time polynomial decay in elastic solids with voids. J. Math. Anal. Appl. 338, 1296–1309 (2008)
Prüss, J.: On the spectrum of C$_0$-semigroups. Trans. AMS 28, 847–857 (1984)
Santos, M.L., Campelo, A.D.S., AlmeidaJúnior, D.S.: On the decay rates of porous elastic systems. J. Elast. 127, 79–101 (2017)
Santos, M.L., Almeida Júnior, D.S.: On porous-elastic system with localized damping. Z. Angew. Math. Phys. 67, 63 (2016). https://doi.org/10.1007/s00033-016-0622-6
Soufyane, A.: Energy decay for porous-thermo-elasticity systems of memory type. Appl. Anal. 87, 451–464 (2008)
Soufyane, A., Afilal, M., Chacha, M.: General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type. Nonlinear Anal. 72, 3903–3910 (2010)
Soufyane, A., Afilal, M., Chacha, M.: Boundary stabilization of memory type for the porous-thermo-elasticity system. Abstr. Appl. Anal. 2009, article number: 280790 (2009)
Xavier Pamplona, P., Muñoz Rivera, J.E., Quintanilla, R.: Stabilization in elastic solids with voids. J. Math. Anal. Appl. 379, 682–705 (2011)
Freitas, M.M., Santos, M.L., Langa, J.A.: Porous elastic system with nonlinear damping and sources terms. J. Differ. Equ. 264(4), 2970–3051 (2018)
Raposo, C.A., Apalara, T.A., Ribeiro, J.O.: Analyticity to transmission problem with delay in porous-elasticity. J. Math. Anal. Appl. 466, 819–834 (2018). https://doi.org/10.1016/j.jmaa.2018.06.017
Barbu, V.: Analysis and Control of Nonlinear Infinite-Dimensional Systems. Academic Press Inc, Boston (1993)
Brézis, H.: Operateurs Maximaux Monotones et Semigroups de Contractions dans les Spaces de Hilbert. North Holland Publishing Co., Amsterdam (1973)
Pazy, A.: semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
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The first author has been partially supported by the CNPq Grant 303026/2018-9. The authors thank for the referee for your careful reading and valuable suggestions.
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Santos, M.L., Almeida Júnior, D.S. & Cordeiro, S.M.S. Energy decay for a porous-elastic system with nonlinear localized damping. Z. Angew. Math. Phys. 73, 7 (2022). https://doi.org/10.1007/s00033-021-01636-1
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DOI: https://doi.org/10.1007/s00033-021-01636-1