In this work, we study the stability of a semilinear porous elastic system under the effect of a locally nonlinear damping acting in the two equations. Employing the nonlinear semigroup theory, we prove the existence and uniqueness of a global solutions for such a problem. The main goal is to show the uniform stability by imposing minimal amount of support for the damping and without any relation on the coefficients. It is worth mentioning that the result obtained here is different from all existing in the literature and improves the decay rate studies for porous elastic materials. The proof of our main stability result relies on refined arguments of microlocal analysis.
Al-GharabliM. M.MessaoudiS. A. (2019). Well-posedness and a general decay for a nonlinear damped porous thermoelastic system with second sound. Georgian Mathematical Journal, 26, 1–11.
2.
Almeida JúniorD. S.SantosM. L. (2016). On porous-elastic system with localized damping. Zeitschrift für Angewandte Mathematik und Physik, 67, 63.
3.
ApalaraT. A. (2019). General decay of solutions in one-dimensional porous-elastic system with memory. Journal of Mathematical Analysis and Applications, 469(2), 457–471.
4.
ApalaraT. A.FengB. (2019). Optimal decay for a porous elasticity system with memory. Journal of Mathematical Analysis and Applications, 470, 1108–1128.
5.
BarbuV. (2010). Nonlinear differential equations of monotone types in Banach spaces. Springer monographs in mathematics. Springer.
6.
BrézisH. (1973). Operateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert. North Holland Publishing Co.
CasasP. S.QuintanillaR. (2005). Exponential decay in one-dimensional porous-thermo-elasticity. Mechanics Research Communications, 32, 652–658.
9.
CavalcantiM. M.Domingos CavalcantiV. N.FukuokaR.PampuA. B.AstudilloM. (2018). Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping. Nonlinearity, 31(9), 4031–4064.
10.
CavalcantiM. M.Domingos CavalcantiV. N.FukuokaR.SorianoJ. A. (2010). Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result. Archive for Rational Mechanics and Analysis, 197, 925–964.
11.
CavalcantiM. M.Domingos CavalcantiV. N.Gonzalez MartinezV. H.PeraltaV. A.VicenteA. (2020). Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping. Journal of Differential Equations, 269, 8212–8268.
12.
CavalcantiM. M.MansouriS.Gonzalez MartinezV. H. (2022). Uniform stabilization for the coupled semi-linear wave and beam equations with distributed nonlinear feedback. Journal of Mathematical Analysis and Applications, 508, 125858.
13.
CowinS. C. (1985). The viscoelastic behavior of linear elastic materials with voids. Journal of Elasticity, 15, 185–191.
14.
CowinS. C.NunziatoW. (1979). A nonlinear theory of elastic materials with voids. Archive for Rational Mechanics and Analysis, 72, 175–201.
15.
CowinS. C.NunziatoJ. W. (1983). Linear elastic materials with voids. Journal of Elasticity, 13, 125–147.
16.
Dos SantosM. J.RaposoC. A.MirandaL. G. R.FengB. (2023). Exponential stability for a nonlinear porous-elastic system with delay. Communications in Mathematics, 31, 359–379.
17.
DuyckaertsT.ZhangX.ZuazuaE. (2008). On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Annales de l’Institut Henri Poincaré–AN, 25, 1–41.
18.
FarehA.MessaoudiS. (2017). Energy decay for a porous thermoelastic system with thermoelasticity of second sound and with a non-necessary positive definite energy. Applied Mathematics and Computation, 293, 493–507.
19.
FengB. (2018). Uniform decay of energy for a porous thermoelasticity system with past history. Applicable Analysis, 97, 210–229.
20.
FengB.Jorge SilvaM. A. (2023). Long-time dynamics of a problem of strain gradient porous elastic theory with nonlinear damping and source terms. Applicable Analysis. https://doi.org/10.1080/00036811.2023.2228815
21.
FreitasM. M. (2020). Pullback attractors for non-autonomous porous elastic system with nonlinear damping and sources terms. Mathematical Methods in the Applied Sciences, 43, 658–681.
22.
FreitasM. M.SantosM. L.LangaJ. A. (2018). Porous elastic system with nonlinear damping and sources terms. Journal of Differential Equations, 264, 2970–3051.
KochH.TataruD. (2005). Dispersive estimates for principally normal pseudo-differential operators. Communications on Pure and Applied Mathematics, 58, 217–284.
25.
LasieckaI.TataruD. (1993). Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differential and Integral Equations, 6, 507–533.
26.
LiuW.ChenD.MessaoudiS. A. (2020). General decay rates for one-dimensional porous-elastic system with memory: The case of non-equal wave speeds. Journal of Mathematical Analysis and Applications, 482, 123552.
27.
MagañaA.QuintanillaR. (2006). On the time decay of solutions in one-dimensional theories of porous materials. International Journal of Solids and Structures, 43, 3414–3427.
28.
MansouriS. (2024a). Asymptotic behavior for a semilinear Bresse system with variable coefficients and locally distributed nonlinear dissipation. Journal of Mathematical Physics, 65, 041512.
29.
MansouriS. (2024b). Uniform stability of a semilinear coupled Timoshenko beam and an elastodynamic system in an inhomogeneous medium. Applicable Analysis, 103(15), 2808–2828.
30.
MansouriS. (2024c). Uniform stability for a semilinear laminated Timoshenko beams posed in inhomogeneous medium with localized nonlinear damping. Journal of Dynamics and Differential Equations. https://doi.org/10.1007/s10884-024-10369-4
31.
MessaoudiS. A.FarehA. (2015). Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete and Continuous Dynamical Systems — Series B (DCDS-B), 20, 599–612.
32.
Muñoz RiveraJ. E.QuintanillaR. (2008). On the time polynomial decay in elastic solids with voids. Journal of Mathematical Analysis and Applications, 338, 1296–1309.
33.
NidZ.FarehA.ApalaraT. A. (2023). On the decay of a porous thermoelasticity type III with constant delay. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 117.
34.
PamplonaP. X.Muñoz RiveraJ. E.QuintanillaR. (2012). Analyticity in porous-thermoelasticity with microtemperatures. Journal of Mathematical Analysis and Applications, 394, 645–655.
RuizA. (1992). Unique continuation for weak solutions of the wave equation plus a potential. Journal de Mathématiques Pures et Appliquées, 71, 455–467.
37.
SantosM. L.CampeloA. D. S.Almeida JúniorD. S. (2017). Rates of decay for porous elastic system weakly dissipative. Acta Applicandae Mathematicae, 151, 1–26.
38.
SoufyaneA. (2008). Energy decay for porous-thermo-elasticity systems of memory type. Applicable Analysis, 87, 451–464.
39.
SoufyaneA.AfilalM.AouamT.ChachaM. (2010). General decay of solutions of a linear one-dimensional porous-thermoelasticity system with a boundary control of memory type. Nonlinear Analysis: Theory, Methods & Applications, 72, 3903–3910.
