This paper investigates a Kirchhoff wave equation with a memory term and a weak damping. First, we establish the global well-posedness of the dynamical system using the semigroup method. Next, we apply the energy method to derive a bounded absorbing set and show that the system is asymptotically smooth. Furthermore, we establish a stabilizability like inequality, which leads to the existence of a global attractor with finite fractal and Hausdorff dimensions.
AndradeD.Jorge SilvaM. A.MaT. F. (2012). Exponential stability for a plate equation with -Laplacian and memory terms. Mathematical Methods in the Applied Sciences, 35(4), 417–426.
2.
ChueshovI.LasieckaI. (2010). Von Karman evolution equations. Springer-Verlag.
3.
DafermosC. M. (1970). Asymptotic stability in viscoelasticity. Archive for Rational Mechanics and Analysis, 37, 297–308.
4.
GiorgiC.GrasseliM.PataV. (2001). Well-posedness and longtime behavior of the phase-field model with memory in a history space setting. Quarterly of Applied Mathematics, 59(4), 701–736.
5.
GiorgiC.MarzocchiA.PataV. (1998). Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Differential Equations and Applications, 5(3), 333–354.
6.
GiorgiC.PataV. (2001). Stability of abstract linear thermoelastic systems with memory. Mathematical Models & Methods in Applied Sciences, 11(4), 627–644.
7.
GrasselliM.PataV. (2002). Uniform attractors of nonautonomous dynamical systems with memory. In A. Lorenzi, & B. Ruf (Eds.), Evolution equations, semigroups and functional analysis. Progress in Nonlinear Differential Equations and Their Applications (vol. 50, pp.155–178). Birkhäuser Basel.
8.
HaleJ. K. (1988). Asymptotic behavior of dissipative systems, mathematical surveys and monographs (vol. 25). American Mathematical Society.
9.
HosoyaM.YamadaY. (1991). On some nonlinear wave equations II: Global existence and energy decay of solutions. Journal of the Faculty of Science, the University of Tokyo. Section 1A, Mathematics, 38, 239–250.
10.
Jorge SilvaM. A.MaT. F. (2013). Long-time dynamics for a class of Kirchhoff models with memory. Journal of Mathematical Physics, 54(2), 021505.
11.
Jorge SilvaM. A.MaT. F. (2013). On a viscoelastic plate equation with history setting and perturbation of -Laplacian type. IMA Journal of Applied Mathematics, 78(6), 1130–1146.
12.
KirchhoffG. (1883). Vorlesungen über mathematische physik. Leipzig.
13.
LvP.LinG. (2023). Exponential attractor for Kirchhoff model with time delay and thermal effect. Results in Applied Mathematics, 19, 100382.
14.
NishiharaK.YamadaY. (1990). On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkcialaj Ekvacioj, 33, 151–159.
15.
OnoK. (1997). Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. Journal of Differential Equations, 137(2), 273–301.
16.
PazyA. (1983). Semigroups of linear operators and applications to partial differential equations (vol. 44). Springer.
17.
QinY. (2008). Nonlinear parabolic–hyperbolic coupled systems and their attractors. Birkhäuser Verlag AG/Springer Science + Business Media.
18.
QinY. (2016). Integral and discrete inequalities and their applications, I. Springer International Publishing.
19.
QinY. (2016). Integral and discrete inequalities and their applications, II. Springer International Publishing.
20.
QinY. (2017). Analytic inequalities and their applications in PDEs. Springer International Publishing.
QinY.WangH. (2024). Global and exponential attractors for viscoelastic Kirchhoff plate equation with memory and time delay. Mathematical Methods in the Applied Sciences, 48(3), 3316–3336.
23.
QinY.WangH.YangB. (2024). Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory term. Asymptotic Analysis, 137(1–2), 85–95.
24.
QinY.YangB. (2024). Attractors for non-classical diffusion equations and Kirchhoff wave equations. EDP Sciences.
25.
SegalI. (1963). Non-linear semigroups. Annals of Mathematics, 78(2), 339–364.
26.
SimonJ. (1987). Compact sets in the space . Annali di Matematica Pura ed Applicata, 146(1), 65–96.
27.
YangZ. (2007). Longtime behavior of the Kirchhoff type equation with strong damping on . Journal of Differential Equations, 242(2), 269–286.
28.
YangB.QinY.MiranvilleA.WangK. (2024). Existence and regularity of global attractors for a Kirchhoff wave equation with strong damping and memory. Nonlinear Analysis: Real World Applications, 79, 104096.
29.
ZhengS (2004). Nonlinear evolution equations, pitman series monographs and survey in pure and applied mathematics (vol. 133). Chapman Hall/CRC Press.
