Bresse Beam with Damping and Logarithmic Source

Abstract

This paper investigates the stabilization of a Bresse system with internal damping and logarithmic source. The authors use the potential well theory. For initial data in the stability set created by the Nehari surface, the existence of a global solution is proved by using Faedo-Galerkin’s approximation. The Nakao theorem gives the exponential decay. A numerical approach is presented to illustrate the results obtained.

Keywords

Bresse beam / Logarithmic source / Global solution / Exponential decay / Numerical approach / 35B40 / 35L70 / 35A01 / 74H40 / 93D15

References

[1] Adams R A. Sobolev Spaces. 1975, New York, Academic Press

[2] Akil M, Badawi H, Nicaise S, Wehbe A. On the stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force. Z. Angew. Math. Phys.. 2021, 72: 1-27

[3] Al-Gharabli M M, Messaoudi S A. Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term. J. Evol. Equ.. 2018, 18: 105-125

[4] Al-Mahdi A M, Al-Gharabli M M, Ali S M. New stability result for a Bresse system with one infinite memory in the shear angle equation. Discrete Contin. Dyn. Syst. S. 2022, 15: 995-1014

[5] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J. Functional Analysis. 1973, 14: 349-381

[6] Barrow J D, Parsons P. Inflationary models with logarithmic potentials. Phys. Rev. D. 1995, 52: 5576-5587

[7] Bekhouche R, Guesmia A, Messaoudi S A. Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements. Arab. J. Math.. 2022, 11: 155-178

[8] Bialynicki-Birula I, Mycielski J. Wave equations with logarithmic nonlinearities. Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys.. 1975, 23: 461-466

[9] Bresse J A C. Cours de Mécanique Appliquée - Résistance des Matériaux et Stabilité des Constructions. 1859, Paris, Gauthier-Villars

[10] Coddington E A, Levinson N. Theory of Ordinary Differential Equations. 1955, New York, McGraw-Hill Inc.

[11] Cordeiro S M S, Pereira D C, Baldez C A C, Raposo C A. Global existence and asymptotic behavior for a Tirnoshenko system with internal damping and logarithmic source terms. Arab. J. Math.. 2023, 12: 105-118

[12] Cordeiro S M S, Pereira D C, Ferreira J, Raposo C A. Global solutions and exponential decay to a Klein-Gordon equation of Kirchhoff-Carrier type with strong damping and nonlinear logarithmic source term. Partial. Differ. Equ. Appl. Math.. 2019, 99: 1-6

[13] Elishakoff I. Handbook on the Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories. 2019, Singapore, World Scientific in press

[14] Elishakoff I. Who developed the so-called Timoshenko beam theory?. Math. Mech. Solids. 2020, 25: 97-116

[15] Elishakoff I. Stepan Prokofievich Timoshenko and America. J. Appl. Math. Mech.. 2021, 3: 1-18

[16] Fatori L H, Rivera J E M. Rates of decay to weak thermoelastic Bresse system. IAM J. Appl. Math.. 2010, 75: 881-904

[17] Hale J K. Ordinary Differential Equations. 19972New York, Dover Publications

[18] Hiramatsu T, Kawasaki M, Takahashi F. Numerical study of Q-ball formation in gravity mediation. J. Cosmol. Astropart. Phys.. 2010, 6: 1-28

[19] Hughes T J R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. 2000, New York, Dover Publications INC.

[20] Lions J L. Quelques méthodes de résolution des problèmes aux limites non linéaires. 1969, Paris, Dunod-Gauthier Villars

[21] Nakao M. Decay of solutions for some nonlinear evolution equations. J. Math. Analysis Appl.. 1977, 60: 542-549

[22] Newmark N M. A method of computation for structural dynamics. J. Engrg. Mech.. 1959, 85: 67-94

[23] Noun N, Wehbe A. Weakly locally internal stabilization of elastic Bresse system. C. R. Math.. 2012, 350: 493-498

[24] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math.. 1975, 22: 273-303

[25] Pereira D C, Cordeiro S M S, Raposo C A, Maranhão C H M. Global existence and uniform decay of solutions for a Kirchhoff beam equation with nonlinear damping and source term. Electron. J. Differ. Equ.. 2021, 21: 1-14

[26] Pereira D C, Raposo C A, Maranhão C H M, Cattai A P. Global existence and uniform decay of solutions for a Kirchhoff beam equation with nonlinear damping and source term. Differ. Equ. Dyn. Syst.. 2021, 2021: 1-14

[27] Prathap G, Bhashyam G R. Reduced integration and the shear-flexible beam element. Internat. J. Numer. Methods Engrg.. 1982, 18: 195-210

[28] Rankine W Y M. A manual of applied mechanics. 1858, London, R. Griffin and Co Ltd.: 342344

[29] Rayleigh L. The Theory of Sound. 1945, New York, Dover

[30] Sattinger D H. On global solution of nonlinear hyperbolic equations. Arch. Rational Mech. Anal.. 1968, 30: 148-172

[31] Timoshenko S P. A course of elasticity theory, Part 2: Rods and plates. 1916, St. Petersburg, A. E. Collins Publishers

[32] Timoshenko S P. On the correction for shear of the differential equation for transverse vibrations of prismatic bar. Philos. Mag.. 1921, 41: 744-746

[33] Willem M. Minimax Theorems. 1996, Boston, Birkhäuser

[34] Ye Y. Global solution and blow-up of logarithmic Klein-Gordon equation. Bull. Korean Math. Soc.. 2020, 57: 281-294

[35] Zloshchastiev K G. Logarithmic Nonlinearity in Theories of Quantum Gravity: Origin of Time and Observational Consequences. Gravit. Cosmol. 2019, 16: 288-297