Global existence and decay of solutions of a singular nonlocal viscoelastic system

  • Alaeddin Draifia
  • Abderrahmane Zarai
  • Salah Boulaaras


In this work, we consider a singular one-dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition. By using the potential well theory we obtain the existence of a global solution. Then, we prove the general decay result, by constructing Lyapunov functional and using the perturbed energy method.


Viscoelastic equations Global existence General decay 

Mathematics Subject Classification

35L35 35L20 



The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions which helped them to improve the paper.


  1. 1.
    Cahlon, D.M., Shi, P.: Stepwise stability for the heat equation with a nonlocal constraint. SIAM. J. Numer. Anal. 32, 571–593 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cannon, R.: The solution of heat equation subject to the specification of energy. Q. Appl. Math. 21, 155–160 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Capasso, V., Kunisch, K.: A reaction-diffusion system arising in modeling man-environment diseases. Q. Appl. Math. 46, 431–449 (1988)CrossRefzbMATHGoogle Scholar
  4. 4.
    Choi, Y.S., Chan, K.Y.: A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Anal. 18, 317–331 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chunlai, M., Jie, M.: On a system of nonlinear wave equations with Balakrishnan-Taylor damping. Z. Angew. Math. Phys. 65(1), 91–113 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ewing, R.E., Lin, T.: A class of parameter estimation techniques for fluid flow in porous media. Adv. Water Resour. 14, 89–97 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ionkin, N.I., Moiseev, E.I.: A problem for the heat conduction equation with two-point boundary condition. Differ. Uravn. 15(7), 1284–1295 (1979)zbMATHGoogle Scholar
  8. 8.
    Ionkin, N.I.: Solution of boundary value problem in heat conduction theory with nonclassical boundary conditions. Differ. Uravn. 13(2), 1177–1182 (1977)Google Scholar
  9. 9.
    Kartynnik, A.V.: Three-point boundary value problem with an integral space-variable condition for a second order parabolic equation. Differ. Eq. 26, 1160–1162 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kamynin, L. I.: A boundary-value problem in the theory of heat conduction with non-classical boundary conditions. (Russian) Ž. Vyč isl. Mat. i Mat. Fiz. 4, 1006–1024 (1964)Google Scholar
  11. 11.
    Li, M.R., Tsai, L.Y.: Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Anal. Theory Methods Appl. 54, 1397–1415 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kafini, M., Messaoudi, S.: A blow up result for a viscoelastic system in \({\mathbb{R}}^{n}\). Electron. J. Differ. Eq. 113, 7 (2003)zbMATHGoogle Scholar
  13. 13.
    Mesloub, S., Bouziani, A.: Problème mixte avec conditions aux limites intégrales pour une classe d’équations paraboliques bidimensionnelles. Bull. Classe Sci. Acad. R. Belg. 6, 59–69 (1998)zbMATHGoogle Scholar
  14. 14.
    Mesloub, S., Bouziani, A.: Mixed problem with a weighted integral condition for a parabolic equation with Bessel operator. J. Appl. Math. Stoch. Anal. 15(3), 291–300 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mesloub, S., Messaoudi, S.A.: Global existence, decay, and blow up of solutions of a singular nonlocal viscoelastic problem. Acta Appl. Math. 110, 705–724 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mesloub, S., Messaoudi, S.A.: A non local mixed semilinear problem for second order hyperbolic equations. Electron. J. Differ. Eq. 2003(30), 1–17 (2003)zbMATHGoogle Scholar
  17. 17.
    Mesloub, S., Lekrine, N.: On a nonlocal hyperbolic mixed problem. Acta Sci. Math. (Szeged) 70, 65–75 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mesloub, S.: On a singular two dimensional nonlinear evolution equation with non local conditions. Nonlinear Anal. 68, 2594–2607 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mesloub, S.: A nonlinear nonlocal mixed problem for a second order parabolic equation. J. Math. Anal. Appl. 316, 189–209 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mesloub, S., Mesloub, F.: Solvability of a mixed nonlocal problem for a nonlinear singular viscoelastic equation. Acta. Appl. Math. 110, 109–129 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Messaoudi, S.A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58–66 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pulkina, L.S.: A nonlocal problem with integral conditions for hyperbolic equations. Electron. J. Differ. Eq. 45, 1–6 (1999)Google Scholar
  23. 23.
    Pulkina, L.S.: On solvability in \(\text{ L }^{2}\) of nonlocal problem with integral conditions for a hyperbolic equation. Differ. Uravn. 36(2), 316 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shi, P., Shilor, M.: Design of Contact Patterns in One Dimensional Thermoelasticity, in Theoretical Aspects of Industrial Design. SIAM, Philadelphia (1992)Google Scholar
  25. 25.
    Shi, P.: Weak solution to an evolution problem with a non local constraint. SIAM J. Math. Anal. 24(1), 46–58 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wu, S.: Blow-up of solutions for a singular nonlocal viscoelastic equation. J. Partial Differ. Eq. 24(2), 140–149 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yurchuk, N.I.: Mixed problem with an integral condition for certain parabolic equations. Differ. Uravn. 22(19), 2117–2126 (1986)Google Scholar
  28. 28.
    Zaraï, A., Tatar, N., Abdelmalek, S.: Elastic membrane equation with memory term and nonlinear boundary damping: global existence, decay and blowup of the solution. Acta Math. Sci. Ser. B Engl. Ed. 33(1), 84–106 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zarai, A., Draifia, A., Boulaaras, S.: Blow up of solutions for a system of nonlocal singular viscoelatic equations. Appl Anal. 97, 2231–2245 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Boulaaras, S., 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).
  31. 31.
    Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Correa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of \(p\)-Kirchhoff type via variational methods. Bull. Aust. Math. Soc. 74, 263–277 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Correa, F.J.S.A., Figueiredo, G.M.: On a \(p\)-Kirchhoff equation type via Krasnoselkii’s genus. Appl. Math. Lett. 22, 819–822 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hai, D.D., Shivaji, R.: An existence result on positive solutions for a class of \(p-\)Laplacian systems. Nonlinear Anal. 56, 1007–1010 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Guefaifia, R., Boulaaras, S.: Existence of positive radial solutions for (p(x), q(x))-Laplacian systems. Appl. Math. E-Notes 18, 209–218 (2018)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Ghfaifia, R., Boulaaras, S.: Existence of positive solution for a class of (p(x), q(x))-Laplacian systems. Rend. Circ. Mat. Palermo, II. Ser. 67, 93–103 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • Alaeddin Draifia
    • 1
  • Abderrahmane Zarai
    • 1
  • Salah Boulaaras
    • 2
    • 3
  1. 1.Laboratory of Mathematics, Informatics and Systems (LAMIS), Department of Mathematics and Computer ScienceLarbi Tebessi UniversityTebessaAlgeria
  2. 2.Department of Mathematics, College of Sciences and Arts, Al-RassQassim UniversityBuraidahKingdom of Saudi Arabia
  3. 3.Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO)University of Oran 1OranAlgeria

Personalised recommendations