Global well-posedness and exponential stability for Maxwell’s equations under delayed boundary condition in metamaterials. (English) Zbl 1518.35584
Summary: We develop an initial-boundary value problem derived from the Maxwell’s system with a nonlinear feedback-type boundary mechanism in metamaterials, which both involves polarization, magnetization effect and time-localized delay effect in a bounded domain. Based on the nonlinear semigroup theory and the properties of viscoelasticity theory, we show the well-posedness of solution in an appropriate Hilbert space. Under some suitable assumptions and geometric conditions, we prove the exponential stability of the Maxwell’s system.
MSC:
| 35Q61 | Maxwell equations |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 68U07 | Computer science aspects of computer-aided design |
| 93B52 | Feedback control |
| 93D15 | Stabilization of systems by feedback |
| 35R07 | PDEs on time scales |
Keywords:
Maxwell’s equations; nonlinear boundary condition; delay feedback; well-posedness; exponential stabilityReferences:
| [1] | Basharin, A.; Kafesaki, M.; Economou, E.; Soukoulis, C., Backward wave radiation from negative permittivity waveguides and its use for THz sub-wavelength imaging, Opt. Express, 20, 12752-12760 (2012) |
| [2] | Watts, C.; Liu, X.; Padilla, W., Metamaterial electromagnetic wave absorbers, Adv. Mater., 24, 98-120 (2012) |
| [3] | Zhang, S.; Xia, C.; Fang, N., Broadband acoustic cloak for ultrasound waves, Phys. Rev. Lett., 106, Article 024301 pp. (2011) |
| [4] | Chen, H.; Chen, M., Flipping photons backward: reversed Cherenkov radiation, Mater. Today, 14, 34-41 (2011) |
| [5] | Duan, Z.; Wu, B.; Xi, S.; Chen, H.; Chen, M., Research progress in reversed Cherenkov radiation in double-negative metamaterials, Prog. Electromagn. Res., 90, 75-87 (2009) |
| [6] | Xie, Z.; Wang, J.; Wang, B.; Chen, C., Solving Maxwell’s equation in meta-materials by a CG-DG method, Commun. Comput. Phys., 19, 1242-1264 (2016) · Zbl 1388.65111 |
| [7] | Shi, C.; Li, J.; Shu, C., Discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials on unstructured meshes, J. Comput. Appl. Math., 342, 147-163 (2018) · Zbl 1462.65156 |
| [8] | Li, J.; Huang, Y., Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials (2013), Springer: Springer Berlin, Heidelberg · Zbl 1304.78002 |
| [9] | Li, J.; Fang, Z.; Lin, G., Finite element analysis for wave propagation in double negative metamaterials, Comput. Methods Appl. Mech. Eng., 335, 24-51 (2007) · Zbl 1440.78007 |
| [10] | Cagnol, J.; Eller, M., Boundary regularity for Maxwell’s equations with applications to shape optimization, J. Differ. Equ., 250, 1114-1136 (2011) · Zbl 1208.35144 |
| [11] | Benaissa, A.; Benguessoum, A., Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback, Int. J. Dyn. Syst. Differ. Equ., 5, 1-26 (2014) · Zbl 1331.35356 |
| [12] | Eller, M.; Lagnese, J.; Nicaise, S., Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math., 21, 135-165 (2002) · Zbl 1119.93402 |
| [13] | Zemanová, V., Quasi-static Maxwell’s equations with a dissipative non-linear boundary condition: full discretization, J. Math. Anal. Appl., 418, 31-46 (2014) · Zbl 1312.78001 |
| [14] | Nicaise, S., Exact boundary controllability of Maxwell’s equations in heterogeneous media and application to an inverse source problem, SIAM J. Control Optim., 38, 1145-1170 (2000) · Zbl 0963.93041 |
| [15] | Nicaise, S.; Pignotti, C., Boundary stabilization of Maxwell’s equations with space-time variable coefficients, ESAIM Control Optim. Calc. Var., 9, 563-578 (2003) · Zbl 1063.93041 |
| [16] | Kirane, M.; Said-Houari, B., Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62, 1065-1082 (2011) · Zbl 1242.35163 |
| [17] | Bahlil, M., On convexity for energy decay rates of a viscoelastic equation with a dynamic boundary and nonlinear delay term in the nonlinear internal feedback, Palest. J. Math., 7, 559-578 (2018) · Zbl 1393.35009 |
| [18] | Nibbi, R.; Polidoro, S., Exponential decay for Maxwell equations with a boundary memory condition, J. Math. Anal. Appl., 302, 30-55 (2005) · Zbl 1067.35121 |
| [19] | Shaw, S., Finite element approximation of Maxwell’s equations with Debye memory, Adv. Numer. Anal., 2010, 1-28 (2010) · Zbl 1221.78049 |
| [20] | Golden, J.; Graham, G., Boundary Value Problems in Linear Viscoelasticity (1988), Springer: Springer Berlin, Heidelberg · Zbl 0655.73021 |
| [21] | Giuseppe, T., Heat equation and convolution inequalities, Milan J. Math., 82, 183-212 (2014) · Zbl 1303.26023 |
| [22] | Lagnese, J., Exact boundary controllability of Maxwell’s equations in a general region, SIAM J. Control Optim., 27, 374-388 (1989) · Zbl 0678.49032 |
| [23] | Khusainov, D.; Pokojovy, M.; Racke, R., Strong and mild extrapolated \(L^2\)-solutions to the heat equation with constant delay, SIAM J. Math. Anal., 47, 427-454 (2015) · Zbl 1331.35357 |
| [24] | Showalter, R., Monotone operators in Banach space and nonlinear partial differential equations, Math. Surv. Monogr. (1997) · Zbl 0870.35004 |
| [25] | Eller, M.; Lagnese, J.; Nicaise, S., Stabilization of heterogeneous Maxwell’s equations by linear and nonlinear boundary feedbacks, Electron. J. Differ. Equ., 2002, 1-26 (2002) · Zbl 1030.93026 |
| [26] | Anikushyn, A.; Demchenko, A.; Pokojovy, M., On a Kelvin-Voigt viscoelastic wave equation with strong delay, SIAM J. Math. Anal., 51, 4382-4412 (2019) · Zbl 1427.35271 |
| [27] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2010), Springer: Springer New York · Zbl 1218.46002 |
| [28] | Costabel, M.; Dauge, M., Un résultat de densité pour les équations de Maxwell régularisées dans un domaine lipschitzien, C. R. Acad. Sci., 327, 849-854 (1998) · Zbl 0921.35169 |
| [29] | Barbu, V., Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics (2010) · Zbl 1197.35002 |
| [30] | Vargas-De-León, C., Lyapunov functionals for global stability of Lotka-Volterra cooperative systems with discrete delays, (Abstraction and Application, vol. 12 (2015)), 42-50 |
| [31] | Anikushyn, A.; Pokojovy, M., Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell’s equations under a nonlinear boundary feedback with delay, J. Comput. Appl. Math., 475, 278-312 (2019) · Zbl 1416.35258 |
| [32] | Eller, M., Continuous observability for the anisotropic Maxwell system, Appl. Math. Optim., 55, 185-201 (2007) · Zbl 1128.35100 |
| [33] | Yao, C., The solvability of coupling the thermal effect and magnetohydrodynamics field with turbulent convection zone and the flow field, J. Math. Anal. Appl., 476, 495-521 (2019) · Zbl 1448.76199 |
| [34] | Huang, Q.; Jia, S.; Xu, F.; Xu, Z.; Yao, C., Solvability of wave propagation with Debye polarization in nonlinear dielectric materials and its finite element methods approximation, Appl. Numer. Math., 146, 145-164 (2019) · Zbl 1477.78002 |
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