Decay estimate and global existence of a semilinear Mindlin-Timoshenko plate system with full frictional damping in the whole space

Li, Kaiqiang; Xue, Rui

doi:10.1090/qam/1569

Authors: Kaiqiang Li and Rui Xue
Journal: Quart. Appl. Math. 78 (2020), 703-724
MSC (2010): Primary 35A01; Secondary 74H40, 74K10
DOI: https://doi.org/10.1090/qam/1569
Published electronically: March 23, 2020
MathSciNet review: 4148824
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we are interested in a semilinear Mindlin-Timoshenko system with an irrotationality condition for the angle variables. Under the smallness assumption on the initial data, the decay rate of solutions is obtained by using both the multiplier method in the Fourier space and the fundamental solution method. Moreover, we also prove the global existence of solutions by the standard method.

References

F. D. Araruna and E. Zuazua, Controllability of the Kirchhoff system for beams as a limit of the Mindlin-Timoshenko system, SIAM J. Control Optim. 47 (2008), no. 4, 1909–1938. MR 2421335, DOI https://doi.org/10.1137/060659934
F. D. Araruna, P. Braz E Silva, and E. Zuazua, Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system, J. Syst. Sci. Complex. 23 (2010), no. 3, 414–430. MR 2679535, DOI https://doi.org/10.1007/s11424-010-0137-8
Igor Chueshov and Irena Lasiecka, Global attractors for Mindlin-Timoshenko plates and for their Kirchhoff limits, Milan J. Math. 74 (2006), 117–138. MR 2278731, DOI https://doi.org/10.1007/s00032-006-0050-8
Hugo D. Fernández Sare, On the stability of Mindlin-Timoshenko plates, Quart. Appl. Math. 67 (2009), no. 2, 249–263. MR 2514634, DOI https://doi.org/10.1090/S0033-569X-09-01110-2
Claudio Giorgi and Federico M. Vegni, The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels, J. Math. Anal. Appl. 326 (2007), no. 2, 754–771. MR 2280942, DOI https://doi.org/10.1016/j.jmaa.2006.03.024
Marié Grobbelaar-Van Dalsen, Strong stabilization of models incorporating the thermoelastic Reissner-Mindlin plate equations with second sound, Appl. Anal. 90 (2011), no. 9, 1419–1449. MR 2819989, DOI https://doi.org/10.1080/00036811.2010.530259
Marié Grobbelaar-Van Dalsen, On the dissipative effect of a magnetic field in a Mindlin-Timoshenko plate model, Z. Angew. Math. Phys. 63 (2012), no. 6, 1047–1065. MR 3000714, DOI https://doi.org/10.1007/s00033-012-0206-z
Marié Grobbelaar-Van Dalsen, Stabilization of a thermoelastic Mindlin-Timoshenko plate model revisited, Z. Angew. Math. Phys. 64 (2013), no. 4, 1305–1325. MR 3085916, DOI https://doi.org/10.1007/s00033-012-0289-6
Marié Grobbelaar-Van Dalsen, Exponential stabilization of magnetoelastic waves in a Mindlin-Timoshenko plate by localized internal damping, Z. Angew. Math. Phys. 66 (2015), no. 4, 1751–1776. MR 3377713, DOI https://doi.org/10.1007/s00033-015-0507-0
Marié Grobbelaar-Van Dalsen, Polynomial decay rate of a thermoelastic Mindlin-Timoshenko plate model with Dirichlet boundary conditions, Z. Angew. Math. Phys. 66 (2015), no. 1, 113–128. MR 3304709, DOI https://doi.org/10.1007/s00033-013-0391-4
Yan Guo and Yanjin Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations 37 (2012), no. 12, 2165–2208. MR 3005540, DOI https://doi.org/10.1080/03605302.2012.696296
Kentaro Ide, Kazuo Haramoto, and Shuichi Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 647–667. MR 2413032, DOI https://doi.org/10.1142/S0218202508002802
John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153
Ta-Tsien Li and Yun Mei Chen, Global classical solutions for nonlinear evolution equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 45, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1172318
Akitaka Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 1, 169–189. MR 0420031, DOI https://doi.org/10.2977/prims/1195190962
Maria Grazia Naso and Federico M. Vegni, Asymptotic behavior of the energy to a thermo-viscoelastic Mindlin-Timoshenko plate with memory, Int. J. Pure Appl. Math. 21 (2005), no. 2, 175–198. MR 2153527
B. G. Pachpatte, Inequalities for differential and integral equations, Mathematics in Science and Engineering, vol. 197, Academic Press, Inc., San Diego, CA, 1998. MR 1487077
Michael Pedersen, The functional analytic setting of HUM. II. The Mindlin-Timoshenko plate model, Int. J. Pure Appl. Math. 40 (2007), no. 3, 373–397. MR 2362751
Michael Pedersen, Well posedness and regularity of the controlled Mindlin-Timoshenko plate model, Int. J. Pure Appl. Math. 40 (2007), no. 2, 273–289. MR 2361656
Pei Pei, Mohammad A. Rammaha, and Daniel Toundykov, Global well-posedness and stability of semilinear Mindlin-Timoshenko system, J. Math. Anal. Appl. 418 (2014), no. 2, 535–568. MR 3206668, DOI https://doi.org/10.1016/j.jmaa.2014.03.014
Pei Pei, Mohammad A. Rammaha, and Daniel Toundykov, Local and global well-posedness of semilinear Reissner-Mindlin-Timoshenko plate equations, Nonlinear Anal. 105 (2014), 62–85. MR 3200741, DOI https://doi.org/10.1016/j.na.2014.03.024
Michael Pokojovy, On stability of hyperbolic thermoelastic Reissner-Mindlin-Timoshenko plates, Math. Methods Appl. Sci. 38 (2015), no. 7, 1225–1246. MR 3338151, DOI https://doi.org/10.1002/mma.3140
Reinhard Racke and Belkacem Said-Houari, Decay rates and global existence for semilinear dissipative Timoshenko systems, Quart. Appl. Math. 71 (2013), no. 2, 229–266. MR 3087421, DOI https://doi.org/10.1090/S0033-569X-2012-01280-8
Reinhard Racke, Lectures on nonlinear evolution equations, 2nd ed., Birkhäuser/Springer, Cham, 2015. Initial value problems. MR 3381126
Reinhard Racke, Weike Wang, and Rui Xue, Optimal decay rates and global existence for a semilinear Timoshenko system with two damping effects, Math. Methods Appl. Sci. 40 (2017), no. 1, 210–222. MR 3583048, DOI https://doi.org/10.1002/mma.3983
M. A. Jorge Silva, T. F. Ma, and J. E. Muñoz Rivera, Mindlin-Timoshenko systems with Kelvin-Voigt: analyticity and optimal decay rates, J. Math. Anal. Appl. 417 (2014), no. 1, 164–179. MR 3191419, DOI https://doi.org/10.1016/j.jmaa.2014.02.066

W. K. Wang and R. Xue, Decay estimate and global existence of semilinear thermoelastic Timoshenko system with two damping effects, Acta Math. Sci. 39B(6) (2019), 1461-1486.