Asymptotic behavior of a viscoelastic wave equation with a delay in internal fractional feedback

• Autorzy: Aounallah Radhouane , Choucha Abdelbaki , Boulaaras Salah , Zarai Abderrahmane

Identyfikatory

DOI

10.24425/acs.2024.149665

• Języki publikacji: EN

Abstrakty

EN

We consider the viscoelastic wave equation with a time delay term in internal fractional feedback. By employing the energy method along with the Faedo-Galerkin procedure, we establish the global existence of solutions, subject to certain conditions. Additionally, we demonstrate how appropriate Lyapunov functionals can lead to general decay results of the energy.

Słowa kluczowe

EN

global existence; general decay; relaxation function; delay fractional feedback; partial differential equations

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2024
• Tom: Vol. 34, No. 2
• Strony: 379--413
• Opis fizyczny: Bibliogr. 31 poz., wzory

Twórcy

autor

Aounallah Radhouane

• Department of Mathematics, University of Djillali-Liabes, Sidi-Bell Abbas, Algeria

• https://orcid.org/0000-0003-1357-8728

autor

Choucha Abdelbaki

• Department of Material Sciences, Faculty of Sciences, Amar Teleji Laghouat University, Laghouat, Algeria
• Laboratory of Mathematics and Applied Sciences, Ghardaia University, Ghardaia, Algeria

• https://orcid.org/0000-0002-0969-8519

autor

Boulaaras Salah

• Department of Mathematics, College of Science, Qassim University, 51452, Buraydah, Saudi Arabia

• https://orcid.org/0000-0002-3458-2403

autor

Zarai Abderrahmane

• Laboratory of Mathematics, Informatics and Systems (LAMIS), Department of Mathematics and Computer Science, Larbi Tebessi University, 12002, Tebessa, Algeria

• https://orcid.org/0000-0002-1737-4899

Bibliografia

• [1] M. Aassila, M.M. Cavalcanti and J.A. Soriano: Asymptotic stability and energy Decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM Journal on Control and Optimization, 35(5), (2000) 1581-1602. DOI: 10.1137/S0363012998344981
• [2] R. Aounallah, A. Benaissa and A. Zarai: Blow-up and asymptotic behavior for a wave equation with a time delay condition of fractional type. In: Rendiconti del Circolo Matematico di Palermo Series 2, 70 (2021), 1061-1081. DOI: 10.1007/s12215-020-00545-y
• [3] R. Aounallah, S. Boulaaras, A. Zarai and B. Cherif: General decay and blow up of solution for a nonlinear wave equation with a fractional boundary damping. Mathematical Methods in the Applied Sciences, 43(12), (2020), 7175-7193. DOI: 10.1002/mma.6455
• [4] R. Aounallah, A. Zarai and A. Benaissa: Blow-up of solutions for elastic membranę equations with fractional boundary damping. Communications in Optimization Theory, 2020 2020. DOI: 10.23952/cot.2020.10
• [5] A. Benaissa and S. Gaouar: Exponential decay ror the Lamé system with fractional time delays and boundary feedbacks. Applied Mathematics E-Notes, 21 (2021), 705-717.
• [6] J. Barrow and P. Parsons: Inflationary models with logarithmic potentials. Physical Review D, 52 (1995), 5576-5587. DOI: 10.48550/arXiv.astro-ph/9506049
• [7] E. Blanc, G. Chiavassa and B. Lombard: Biot-JKD model: Simulation of 1D transient poro-elastic waves with fractional derivative. Journal of Computational Physics, 237 (2013), 1-20. DOI: 10.1016/j.jcp.2012.12.003
• [8] M.M. Cavalcanti, V.N. Domingos Cavalcanti and J. Ferreira: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Mathematical Methods in the Applied Sciences, 24(14), (2001), 1043-1053. DOI: 10.1002/mma.250; MR1855298
• [9] S. Boulaaras, R. Guefaifia and N. Mezouar: Global existence and decay for a system of two singular one-dimensional nonlinear viscoelastic equations with general source terms. Applicable Analysis, 101(3), (2022), 824-848. DOI: 10.1080/00036811.2020.1760250
• [10] S. Boulaaras and N. Mezouar: Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term nonlocal boundary condition, and localized damping term. Mathematical Methods in the Applied Sciences, 43(10) (2020), 6140-6164. DOI: 10.1002/mma.6361
• [11] H. Dai and H. Zhang: Exponential growth for wave equation with fractional boundary dissipation and boundary source term. Boundary Value Problems, 2014 (2014). DOI: 10.1186/s13661-014-0138-y
• [12] M. Fabrizio and S. Polidoro: Asymptotic decay for some differential systems with fading memory. Applicable Analysis, 81(6), (2002), 1245-1264. DOI: 10.1080/0003681021000035588
• [13] G.Q. Xu, S.P. Yung and L.K. Li: Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, optimisati on and calculus of variations, 12(4), 770-785.
• [14] S. Gala and M.A. Ragusa: Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices. Applicable Analysis, 95(6), (2016), 1271-1279. DOI: 10.1080/00036811.2015.1061122
• [15] S.A. Messaoudi: Blow up of solutions with positive initial energy in a nonlinear viscoelastic wave equations. Journal of Mathematical Analysis and Applications, 320(2), (2006), 902-915. DOI: 10.1016/j.jmaa.2005.07.022
• [16] B. Mbodje: Wave energy decay under fractional derivative controls. IMA J. Math. Control Inform, 23(2), (2006), 237-257. DOI: 10.1093/imamci/dni056
• [17] G. Liu: The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 28(1), (2020), 263-289. DOI: 10.3934/era.2020016
• [18] Y.C. Liu: On potential wells and vacuum isolating of solutions for semilinear wave equations. Journal of Differential Equations, 192(1), (2003), 155-169. DOI: 10.1016/S0022-0396(02)00020-7
• [19] S.A. Messaoudi and N.-E.Tatar: Exponential and polynomial decay for a quasilinear viscoelastic equation. Nonl. Anal, 68(4), (2008), 785-793. DOI: 10.1016/j.na.2006.11.036
• [20] B. Mbodje, Wave energy decay under fractional derivative controls. IMA J. Math. Control Inform., 23 (2006), 237-257.
• [21] S.A. Messaoudi: General decay of solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Analysis, 69(8), (2008), 2589-2598. DOI: 10.1016/j.na.2007.08.035
• [22] J.E. Muñoz Rivera, M. Naso and E. Vuk: Asymptotic behavior of the energy for electromagnetic system with memory. Mathematical Methods in the Applied Sciences, 27(7), (2004), 819-841. DOI: 10.1002/mma.473
• [23] S. Nicaise and C. Pignotti: Stabilization of the wave equation with boundary or internal distributed delay. Diff. Int. Equs, 21(9-10), (2008), 935-958. DOI: 10.57262/die/1356038593
• [24] T.G. Ha and S.-H. Park: Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity. Advances in Continuous and Discrete Models. Theory and Modern Applications, 2020 (2020), DOI: 10.1186/s13662-020-02694-x
• [25] N. Mezouar and S. Boulaaras: Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation. The Bulletin of the Malaysian Mathematical Society Series 2, 43(3), (2020), 725-755. DOI: 10.1007/s40840-018-00708-2
• [26] N. Mezouar, S. Boulaaras and A. Allahem: Global existence of solutions for the Viscoelastic Kirchhoff Equation with logarithmic source terms. Complexity, 2020(4), (2020), DOI: 10.1155/2020/7105387
• [27] J.E. Munoz Rivera, M. Naso and E. Vuk: Asymptotic behavior of the energy for electromagnetic system with memory. Mathematical Methods in the Applied Sciences, 27(7), (2004), 819-841. DOI: 10.1002/mma.473
• [28] E. Piskin and N. Irkilb: Mathematical behaviour of solutions of the Kirchhoff type equation with logarithmic nonlinearity. Third International Conference of Mathematical Sciences (ICMS 2019), AIP Conference Proceedings, 2183(1), (2019). DOI: 10.1063/1.5136208
• [29] A. Vicente: Wave equation with acoustic/memory boundary conditions. Boletim da Sociedade Paranaense de Matemática, 27(1), (2009), 29-39. DOI: 10.5269/bspm.v27i1.9066
• [30] C.Q. Xu, S.P. Yung and L.K. Li: Stabilization of the wave system with input delay in the boundary control. ESAIM: Control Optim. Calc. Var, 12(4), (2006), 770-785. DOI: 10.1051/cocv:2006021
• [31] H.-C. Zhou and B.-Z. Guo: Boundary feedback stabilization for an unstable time fractional reaction-diffusion equation. SIAM Journal on Control and Optimization, 56(1), (2018), 75-101. DOI: 10.1137/15M1048999