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On the General Stability of a Viscoelastic Wave Equation with an Integral Condition

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Abstract

We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results in the literature.

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References

  1. Beilin, S.A. Existence of solutions for one-dimensional wave equations with nonlocal conditions. Elect. J. Diff. Eqs., 2001(76): 1–8 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Beilin, S.A. On a mixed nonlocal problem for a wave equation. Elect. J. Diff. Eqs., 2006(103): 1–10 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Benouar, N.E., Yurchuk, N.I. A mixed problem with an integral conditions for parabolic equations with a Bessel operator. Differential Equations, 27(12): 1482–1487 (1991)

    MathSciNet  Google Scholar 

  4. Berrimi, S., Messaoudi, S.A. Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Elect. J. Diff. Eqs., 2004(88): 1–10 (2004)

    MATH  Google Scholar 

  5. Berrimi, S., Messaoudi, S.A. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Analysis, 64: 2314–2331 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bouziani, A., Merazga N., Benamira, S. Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions. Nonlinear Analysis, 69: 1515–1524 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouziani, A. A mixed problem with only integral boundary conditions for a hyperbolic equations. Inter. J. Math. Math. Sci., 24: 1279–1291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bouziani, A. Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations. Hiroshima Math. J., 27: 373–390 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cabanillas, E.L., Munoz Rivera, J.E. Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels. Comm. Math. Phys., 177: 583–602 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cannon, J.R. The solution of the heat equation subject to the specification of energy. Quart. Appl. Math., 21: 155–160 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cannon, J.R., Esteve, S.P., Van der Hoeck, J.A. Galerkin procedure for the diffusion equation subject to the specification of mass. SIAM J. Numer. Anal., 24: 499–515 (1987)

    Article  MathSciNet  Google Scholar 

  12. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Elect. J. Diff. Eqs., 2002(44): 1–14 (2002)

    MATH  Google Scholar 

  13. Cavalcanti, M.M., Oquendo, H.P. Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim., 42(4): 1310–1324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, Z.J., Liu, W.J., Chen, D.Q. General Decay Rates for a Laminated Beam with Memory. Taiwanese J. Math., 23(5): 1227–1252 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Denche, M., Marhoune, A.L. A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator. Appl. Math. Letters, 13: 85–89 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fabrizio, M., Morro, A. Mathematical Problems in Linear Viscoelasticity. SIAM Studies in Appl. Math., Vol. 12, Philadelphia, 1992.

  17. Kartynnik, A.V. A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic. Diff. Eqs., 26(9): 1160–1166 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Kirichenko, S.V., Pul’kina, L.S. A Problem with Nonlocal Initial Data for One-Dimensional Hyperbolic Equation. Russian Math., (Iz. VUZ), 58(9): 13–21 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, G., Liu, W.J. General decay for a laminated beam with structural damping and memory: the case of non-equal wave speeds. J. Inte. Eqs. Appls., 30(1): 95–116 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, Z., Zheng, S. Semigroups associated with dissipative systems. Chapman & Hall/CRC Research Notes in Math., Vol. 398, Chapman & Hall/CRC, Boca Raton, FL 1999.

    MATH  Google Scholar 

  21. Liu, K., Liu, Z. On the type of C0-semigroups associated with the abstract linear viscoelastic system. Z. Angew Math. Phys., 47(1): 1–15 (1996)

    Article  MathSciNet  Google Scholar 

  22. Liu, W.J. General decay of solutions to a viscoelastic wave equation with nonlinearlocalized damping. Ann. Acad. Sci. Fenn. Math., 34(1): 291–302 (2009)

    MathSciNet  Google Scholar 

  23. Liu, W.J. General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J. Math. Phys., 50(11): 113506–113517 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, W.J., Sun, Y., Li, G. On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with nonlinear source term. Topol. Methods Nonlinear Anal., 49(1): 299–323 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Liu, W.J., Wang, D.H., Chen, D.Q. General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound. J. Therm. Stresses, 41(6): 758–775 (2008)

    Article  Google Scholar 

  26. Marhoune, A.L., Memou, A. Nonlocal singular problem with integral condition for a second-order parabolic equation. Elect. J. Diff. Eqs., 2015(64): 1–9 (2015)

    MATH  Google Scholar 

  27. Mecheri, H., Mesloub, S., Messaoudi, S.A. On solutions of a singular viscoelastic equation with an integral condition. Georgian Math. J., 16(4): 761–778 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Mesloub, S., Bouziani, A. On a class of singular hyperbolic equation. Internat. J. Math. & Math. Sci., 22(3): 511–519 (1999)

    Article  MATH  Google Scholar 

  29. Mesloub, S., Bouziani, A. Mixed problem with integral conditions for a certain class of hyperbolic equations. J. Appl. Math., 1(3): 107–116 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mesloub, S., Messaoudi, S.A. A three-point boundary-value problem for a hyperbolic equation with a non-local condition. Elect. J. Diff. Eqs., 2002(62): 1–13 (2002)

    MATH  Google Scholar 

  31. Mesloub, S., Messaoudi, S.A. A nonlocal mixed semilinear problem for second-order hyperbolic equations. Elect. J. Diff. Eqs., 2003(30): 1–17 (2003)

    MATH  Google Scholar 

  32. Mesloub, S., Mesloub, F. Solvability of a mixed nonlocal problem for a nonlinear singular viscoelastic equation. Acta Appl. Math., 110(1): 109–129 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mesloub, S., Messaoudi, S.A. Global Existence, Decay, and Blow up of Solutions of a Singular Nonlocal Viscoelastic Problem. Acta Appl. Math., 110(2): 705–724 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Messaoudi, S.A., Al-Khulaifi, W. General and Optimal Decay for a Viscoelastic Equation with Boundary Feedback. Topol. Methods Nonlinear Anal., 51(2): 413–427 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Messaoudi, S.A. General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl., 341(2): 1457–1467 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Messaoudi, S.A., Al-Khulaifi, W. General and optimal decay for a quasilinear viscoelastic equation. Appl. Math. Letters, 66: 16–22 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Munoz Rivera, J.E. Asymptotic behavior in linear viscoelasticity. Quart. Appl. Math., 52(4): 628–648 (1994)

    Article  Google Scholar 

  38. Pul’kina, L.S. A nonlocal problem with integral conditions for hyperbolic equations. Elect. J. Diff. Eqs., 1999(45): 1–6 (1999)

    Google Scholar 

  39. Pul’kina, L.S. A mixed problem with integral condition for the hyperbolic equation. Mathematical Notes, 74(3): 411–421 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pul’kina, L.S. Boundary-Value Problems for a Hyperbolic Equation with Nonlocal Conditions of the I and II Kind. Russian Math,(Iz. VUZ)., 56(4): 62–69 (2012)

    Article  MATH  Google Scholar 

  41. Samarskii, A.A. Some Problems of the Theory of Differential Equations. Differentsial’nye Uravneniya, 16(11): 1925–1935 (1980)

    MathSciNet  Google Scholar 

  42. Tatar, N.E. On a non-dissipative Kirchhoff viscoelastic problem. J. Contemp. Math. Anal., 48(6): 285–293 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wu, S. Blow-up of solutions for a singular nonlocal viscoelastic equation. J. Partial Diff. Eqs., 24(2): 140–149 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Yurchuk, N.I. A mixed problem with an integral condition for some parabolic equations (Russian). Differentsial’nye Uravneniya, 22(12): 2117–2126 (1986)

    MathSciNet  Google Scholar 

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Acknowledgments

The authors thank the anonymous referee for his/her careful reading and valuable suggestions. The second author is sponsored by the University of Sharjah, Research group MASEP.

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Authors and Affiliations

  1. Department of Mathematics and LMPA Laboratory, University Mohamed Seddik Ben Yahia-Jijel, Jijel, 18000, Algeria

    Farida Belhannache

  2. Department of Mathematics, University of Sharjah, Sharjah, 27272, UAE

    Salim A. Messaoudi

Authors
  1. Farida Belhannache
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  2. Salim A. Messaoudi
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Correspondence to Farida Belhannache.

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Belhannache, F., Messaoudi, S.A. On the General Stability of a Viscoelastic Wave Equation with an Integral Condition. Acta Math. Appl. Sin. Engl. Ser. 36, 857–869 (2020). https://doi.org/10.1007/s10255-020-0979-3

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  • DOI: https://doi.org/10.1007/s10255-020-0979-3

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