Elastic membrane equation with dynamic boundary conditions and infinite memory
DOI:
https://doi.org/10.5269/bspm.47621Abstract
In this paper, we study the elastic membrane equation with dynamic boundary conditions, source term and a nonlinear weak damping localized on a part of the boundary and past history. Under some appropriate assumptions on the relaxation function the general decay for the energy have been established using the perturbed Lyapunov functionals and some properties of convex functions.
References
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7. Berrimi, S., Messaoudi, S.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. TMA 64, 2314-2331 (2006) https://doi.org/10.1016/j.na.2005.08.015
8. Cavalcanti, M. M., Domingos Cavalcanti, V.N., Prates Filho, J.S., Soriano, J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integr. Equ. 14, 85-116 (2001) https://doi.org/10.5209/rev_REMA.2001.v14.n1.17054
9. Cavalcanti, M. M., Domingos Cavalcanti, V.N., Soriano, J.A.: Exponentian decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1-14 (2002)
10. Cavalcanti, M. M., Oquendo, H.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310-1324 (2003) https://doi.org/10.1137/S0363012902408010
11. Cavalcanti, M. M., Domingos Cavalcanti, V.N., Martinez, P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. TMA 68,177-193 (2008) https://doi.org/10.1016/j.na.2006.10.040
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13. Doronin, G. G., Larkin, N.A.: Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions. Nonlinear Anal. TMA 8, 1119-1134 (2002) https://doi.org/10.1016/S0362-546X(01)00804-5
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16. Feng, Baowei. General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history. Mediterr. J. Math. 15 (2018), no. 3, Art. 103, 17 pp. https://doi.org/10.1007/s00009-018-1154-4
17. Ferhat, M., Hakem, A.: On convexity for energy decay rates of a viscoelastic wave equation with a dynamic boundary and nonlinear delay term. Facta Univ.Ser. Math. Inform. 30, 67-87 (2015)
18. H. Medekhel, S. Boulaaras, Existence of positive solutions for a class of Kirchhoff parabolic systems with multiple parameters Appl. Math. E-Notes, 18(2018), 295-306
19. Ferhat, M., Hakem, A.: Asymptotic behavior for a weak viscoelastic wave equations with a dynamic boundary and time varying delay term. J. Appl. Math.Comput. 51, 509-526 (2016) https://doi.org/10.1007/s12190-015-0917-3
20. Gerbi, S., Said-Houari, B.: Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Adv. Differ.Equ. 13, 1051-1074 (2008)
21. S. Boulaaras , A well-posedness and exponential decay of solutions for a coupled Lam'e system with viscoelastic term and logarithmic source terms, Applicable Analysis (2019) https://doi.org/10.1080/00036811.2019.1648793
22. Gerbi, S., Said-Houari, B.: Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions. Adv. Nonlinear Anal. 2, 163-193 (2013) https://doi.org/10.1515/anona-2012-0027
23. S. Otmani, S. Boulaaras, A.Allahem. The maximum norm analysis of a nonmatching grids method for a class of parabolic $p(x)$-Laplacien equation, Boletim Sociedade Paranaense de Matematica , (2019) https://doi.org/10.5269/bspm.45218
24. Guesmia, A.: Asymototic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 382, 748-760 (2011) https://doi.org/10.1016/j.jmaa.2011.04.079
25. P. Holmes, Bifurcations to divergence and flutter in flow-induced oscillations-a fnite dimensional analysis, Journal of Sound and Vibration, 53(1977), pp. 471-503. https://doi.org/10.1016/0022-460X(77)90521-1
26. P. Holmes, J. E. Marsden, Bifurcation to divergence and flutter flow induced oscillations; an in nite dimensional analysis, Automatica, Vol. 14 (1978). https://doi.org/10.1016/0005-1098(78)90036-5
27. Kafini, M., Mustafa, M. I.: On the stabilization of a non-dissipative Cauchy viscoelastic problem. Mediterr. J. Math. 13, 5163-5176 (2016) https://doi.org/10.1007/s00009-016-0799-0
28. Komornik, V.: Exact Controllability and Stabilization, the Multiplies Method.RMA, vol. 36. Masson, Paris (1994)
29. Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integr. Equ. 6, 507-533(1993)
30. Liu, W. J., Yu, J.: On decay and blow-up of th esolution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Anal. TMA 74,2175-2190 (2011) https://doi.org/10.1016/j.na.2010.11.022
31. Messaoudi, S.A.: Blow up and global existence in nonlinear viscoelastic wave equations. Math. Nachr. 260, 58-66 (2003) https://doi.org/10.1002/mana.200310104
32. S. Boulaaras , A. Zarai and A. Draifia, Galerkin method for nonlocal mixed boundary value problem for the Moore-Gibson-Thompson equation with integral condition, Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.5540
33. Messaoudi, S.A., Tatar, N. E.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 30, 665-680 (2007) https://doi.org/10.1002/mma.804
34. S. Boulaaras, A. Draifia and A. Alnegga, Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel, Symmetry 2019, 11(2), 226; https://doi.org/10.3390/sym11020226
35. Messaoudi, S.A., Said-Houari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J. Math. Anal. Appl. 365, 277-287 (2010) https://doi.org/10.1016/j.jmaa.2009.10.050
36. Munoz Rivera, J.E., Andrade, D.: Exponential decay of non-linear wave equation with a viscoelastic boundary condition. Math. Methods Appl. Sci. 23,41-61 (2000) https://doi.org/10.1002/(SICI)1099-1476(20000110)23:1<41::AID-MMA102>3.0.CO;2-B
37. M. Maizi, S. Boulaaras, M. Haiour, A. Mansour, Existence of positive solutions of Kirchhoff hyperbolic systems with multiple parameters, Boletim Sociedade Paranaense de Matem'atica, (2019), https://doi.org/10.5269/bspm.45418
38. Mustafa, M.I.: Optimal decay rates for the viscoelastic wave equation. Math.Methods Appl. Sci. 41, 192-204 (2018) https://doi.org/10.1002/mma.4604
39. N. Mezouar, S. Boulaaras: Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation, Applicable Analysis, (2018) https://doi.org/10.1080/00036811.2018.1544621
40. N. Mezouar, S. Boulaaras Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation, Bulletin of the Malaysian Mathematical Sciences Society (2018) https://doi.org/10.1007/s40840-018-00708-2
41. N-e. Tatar and A. A. Zaraı, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. Demonstratio Math. 44(2011), 67-90. https://doi.org/10.1515/dema-2013-0297
42. N-e. Tatar and A. Zaraı, On a Kirchhoff equation with Balakrishnan-Taylor damping and source term. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18(2011), 615-627.
43. S. Boulaaras, R.Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Mathematical Methods in the Applied Sciences, Volume 41, Issue 13, 5203-5210 https://doi.org/10.1002/mma.5071
44. S. Boulaaras and R.Guefaifia, S. Kabli: An asymptotic behavior of positive solutions for a new class of elliptic systems involving of (p(x),q(x))-Laplacian systems. Bol. Soc. Mat. Mex. (2017). https://doi.org/10.1007/s40590-017-0184-4
45. R. Guefaifia and S. Boulaaras Existence of positive radial solutions for (p(x),q(x))-Laplacian systems Appl. Math. E-Notes, 18(2018), 209-218
46. R. Guefaifia and S. Boulaaras, Existence of positive solution for a class of (p(x),q(x))-Laplacian systems, Rend. Circ. Mat. Palermo, II. Ser 67 (2018), 93-103 https://doi.org/10.1007/s12215-017-0297-7
47. S. Boulaaras and R. Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Mathematical Methods in the Applied Sciences, Volume 41, Issue 13, 5203-5210 https://doi.org/10.1002/mma.5071
48. S. Boulaaras and Ghfaifia, S. Kabli, An asymptotic behavior of positive solutions for a new class of elliptic systems involving of (p(x),q(x))-Laplacian systems. Bol. Soc. Mat. Mex. (2017). https://doi.org/10.1007/s40590-017-0184-4
2. A. Zaraı and N-e. Tatar, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping. Arch. Math. (Brno) 46(2010), 47-56.
3. A. Zaraı and N-e. Tatar, Non-solvability of Balakrishnan-Taylor equation with memory term in RN. In: Anastassiou G., Duman O. (eds) Advances in Applied Mathematics and Approximation Theory, Springer Proceedings in Mathematics & Statistics, vol 41. Springer, New York, NY 2013. https://doi.org/10.1007/978-1-4614-6393-1_27
4. A. Zaraı and N-e. Tatar and S. Abdelmalek, Elastic membrance equation with memory term and nonlinear boundary damping: global existence, decay and blowup of the solution. Acta Math. Sci. 33B (2013), 84-106. https://doi.org/10.1016/S0252-9602(12)60196-9
5. R. W. Bass and D. Zes, Spillover, nonlinearity and flexible structures, in The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 (ed. L.W.Taylor), 1991, 1-14.
6. Benaissa, A., Ferhat, M.: Stability results for viscoelastic wave equation with dynamic boundary conditions. arXiv : 1801.02988v1
7. Berrimi, S., Messaoudi, S.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. TMA 64, 2314-2331 (2006) https://doi.org/10.1016/j.na.2005.08.015
8. Cavalcanti, M. M., Domingos Cavalcanti, V.N., Prates Filho, J.S., Soriano, J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integr. Equ. 14, 85-116 (2001) https://doi.org/10.5209/rev_REMA.2001.v14.n1.17054
9. Cavalcanti, M. M., Domingos Cavalcanti, V.N., Soriano, J.A.: Exponentian decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 44, 1-14 (2002)
10. Cavalcanti, M. M., Oquendo, H.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310-1324 (2003) https://doi.org/10.1137/S0363012902408010
11. Cavalcanti, M. M., Domingos Cavalcanti, V.N., Martinez, P.: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. TMA 68,177-193 (2008) https://doi.org/10.1016/j.na.2006.10.040
12. Dafermos, C. M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech.Anal. 37, 297-308 (1970) https://doi.org/10.1007/BF00251609
13. Doronin, G. G., Larkin, N.A.: Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions. Nonlinear Anal. TMA 8, 1119-1134 (2002) https://doi.org/10.1016/S0362-546X(01)00804-5
14. E. H. Dowell, Aeroelasticity of plates and shells, Groninger, NL, Noordhoff Int. Publishing Co. (1975).
15. Fabrizio, M., Polidoro, S.: Asymptotic decay for some differential systems with fading memory. Appl. Anal. 81(6), 1245-1264 (2002) https://doi.org/10.1080/0003681021000035588
16. Feng, Baowei. General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history. Mediterr. J. Math. 15 (2018), no. 3, Art. 103, 17 pp. https://doi.org/10.1007/s00009-018-1154-4
17. Ferhat, M., Hakem, A.: On convexity for energy decay rates of a viscoelastic wave equation with a dynamic boundary and nonlinear delay term. Facta Univ.Ser. Math. Inform. 30, 67-87 (2015)
18. H. Medekhel, S. Boulaaras, Existence of positive solutions for a class of Kirchhoff parabolic systems with multiple parameters Appl. Math. E-Notes, 18(2018), 295-306
19. Ferhat, M., Hakem, A.: Asymptotic behavior for a weak viscoelastic wave equations with a dynamic boundary and time varying delay term. J. Appl. Math.Comput. 51, 509-526 (2016) https://doi.org/10.1007/s12190-015-0917-3
20. Gerbi, S., Said-Houari, B.: Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Adv. Differ.Equ. 13, 1051-1074 (2008)
21. S. Boulaaras , A well-posedness and exponential decay of solutions for a coupled Lam'e system with viscoelastic term and logarithmic source terms, Applicable Analysis (2019) https://doi.org/10.1080/00036811.2019.1648793
22. Gerbi, S., Said-Houari, B.: Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions. Adv. Nonlinear Anal. 2, 163-193 (2013) https://doi.org/10.1515/anona-2012-0027
23. S. Otmani, S. Boulaaras, A.Allahem. The maximum norm analysis of a nonmatching grids method for a class of parabolic $p(x)$-Laplacien equation, Boletim Sociedade Paranaense de Matematica , (2019) https://doi.org/10.5269/bspm.45218
24. Guesmia, A.: Asymototic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 382, 748-760 (2011) https://doi.org/10.1016/j.jmaa.2011.04.079
25. P. Holmes, Bifurcations to divergence and flutter in flow-induced oscillations-a fnite dimensional analysis, Journal of Sound and Vibration, 53(1977), pp. 471-503. https://doi.org/10.1016/0022-460X(77)90521-1
26. P. Holmes, J. E. Marsden, Bifurcation to divergence and flutter flow induced oscillations; an in nite dimensional analysis, Automatica, Vol. 14 (1978). https://doi.org/10.1016/0005-1098(78)90036-5
27. Kafini, M., Mustafa, M. I.: On the stabilization of a non-dissipative Cauchy viscoelastic problem. Mediterr. J. Math. 13, 5163-5176 (2016) https://doi.org/10.1007/s00009-016-0799-0
28. Komornik, V.: Exact Controllability and Stabilization, the Multiplies Method.RMA, vol. 36. Masson, Paris (1994)
29. Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integr. Equ. 6, 507-533(1993)
30. Liu, W. J., Yu, J.: On decay and blow-up of th esolution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Anal. TMA 74,2175-2190 (2011) https://doi.org/10.1016/j.na.2010.11.022
31. Messaoudi, S.A.: Blow up and global existence in nonlinear viscoelastic wave equations. Math. Nachr. 260, 58-66 (2003) https://doi.org/10.1002/mana.200310104
32. S. Boulaaras , A. Zarai and A. Draifia, Galerkin method for nonlocal mixed boundary value problem for the Moore-Gibson-Thompson equation with integral condition, Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.5540
33. Messaoudi, S.A., Tatar, N. E.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Methods Appl. Sci. 30, 665-680 (2007) https://doi.org/10.1002/mma.804
34. S. Boulaaras, A. Draifia and A. Alnegga, Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel, Symmetry 2019, 11(2), 226; https://doi.org/10.3390/sym11020226
35. Messaoudi, S.A., Said-Houari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J. Math. Anal. Appl. 365, 277-287 (2010) https://doi.org/10.1016/j.jmaa.2009.10.050
36. Munoz Rivera, J.E., Andrade, D.: Exponential decay of non-linear wave equation with a viscoelastic boundary condition. Math. Methods Appl. Sci. 23,41-61 (2000) https://doi.org/10.1002/(SICI)1099-1476(20000110)23:1<41::AID-MMA102>3.0.CO;2-B
37. M. Maizi, S. Boulaaras, M. Haiour, A. Mansour, Existence of positive solutions of Kirchhoff hyperbolic systems with multiple parameters, Boletim Sociedade Paranaense de Matem'atica, (2019), https://doi.org/10.5269/bspm.45418
38. Mustafa, M.I.: Optimal decay rates for the viscoelastic wave equation. Math.Methods Appl. Sci. 41, 192-204 (2018) https://doi.org/10.1002/mma.4604
39. N. Mezouar, S. Boulaaras: Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation, Applicable Analysis, (2018) https://doi.org/10.1080/00036811.2018.1544621
40. N. Mezouar, S. Boulaaras Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation, Bulletin of the Malaysian Mathematical Sciences Society (2018) https://doi.org/10.1007/s40840-018-00708-2
41. N-e. Tatar and A. A. Zaraı, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. Demonstratio Math. 44(2011), 67-90. https://doi.org/10.1515/dema-2013-0297
42. N-e. Tatar and A. Zaraı, On a Kirchhoff equation with Balakrishnan-Taylor damping and source term. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18(2011), 615-627.
43. S. Boulaaras, R.Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Mathematical Methods in the Applied Sciences, Volume 41, Issue 13, 5203-5210 https://doi.org/10.1002/mma.5071
44. S. Boulaaras and R.Guefaifia, S. Kabli: An asymptotic behavior of positive solutions for a new class of elliptic systems involving of (p(x),q(x))-Laplacian systems. Bol. Soc. Mat. Mex. (2017). https://doi.org/10.1007/s40590-017-0184-4
45. R. Guefaifia and S. Boulaaras Existence of positive radial solutions for (p(x),q(x))-Laplacian systems Appl. Math. E-Notes, 18(2018), 209-218
46. R. Guefaifia and S. Boulaaras, Existence of positive solution for a class of (p(x),q(x))-Laplacian systems, Rend. Circ. Mat. Palermo, II. Ser 67 (2018), 93-103 https://doi.org/10.1007/s12215-017-0297-7
47. S. Boulaaras and R. Guefaifia, Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters, Mathematical Methods in the Applied Sciences, Volume 41, Issue 13, 5203-5210 https://doi.org/10.1002/mma.5071
48. S. Boulaaras and Ghfaifia, S. Kabli, An asymptotic behavior of positive solutions for a new class of elliptic systems involving of (p(x),q(x))-Laplacian systems. Bol. Soc. Mat. Mex. (2017). https://doi.org/10.1007/s40590-017-0184-4
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