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Arabian Journal of Mathematics

, Volume 7, Issue 2, pp 113–145 | Cite as

On wave equation: review and recent results

  • Salim A. Messaoudi
  • Ala A. Talahmeh
Open Access
Article

Abstract

The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs.

Mathematics Subject Classification

35L05 35L70 35B44 35B35 

Notes

References

  1. 1.
    Alabau-Boussouira, F.: Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51(1), 61–105 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Al’shin, A.B.; Korpusov, M.O.; Sveshnikov, A.G.: Blow-up in nonlinear Sobolev type equations. In: Series in Nonlinear Analysis and Applications, vol. 15, 660 p. Walter de Gruyter (2011)Google Scholar
  3. 3.
    Antontsev, S.; Zhikov, V.: Higher integrability for parabolic equations of \(p(x, t)\)-Laplacian type. Adv. Differ. Equ. 10(9), 1053–1080 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Antontsev, S.; Shmarev, S.: Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math. 234(9), 2633–2645 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Antontsev, S.: Wave equation with \(p(x, t)\)-Laplacian and damping term: existence and blow-up. Differ. Equ. Appl. 3(4), 503–525 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Antontsev, S.: Wave equation with \(p(x, t)\)-Laplacian and damping term: blow-up of solutions. C. R. Mec. 339(12), 751–755 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Antontsev, S.; Ferreira, J.: Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions. Nonlinear Anal. Theory Methods Appl. 93, 62–77 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Antontsev, S.; Shmarev, S.: Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blow-up. In: Atlantis Studies in Differential Equations, vol. 4. Atlantis Press (2015)Google Scholar
  9. 9.
    Autuori, G.; Pucci, P.; Salvatori, M.: Global nonexistence for nonlinear Kirchhoff systems. Arch. Ration. Mech. Anal. 196(2), 489–516 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ball, J.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. 28(4), 473–486 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Benaissa, A.; Messaoudi, S.A.: Blow up of solutions of a nonlinear wave equation. J. Appl. Math. 2(2), 105–108 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Benaissa, A.; Messaoudi, S.A.: Blow-up of solutions for Kirchhoff equation of \(q\)-Laplacian type with nonlinear dissipation. Colloq. Math. 94(1), 103–109 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Benaissa, A.; Messaoudi, S.A.: Blow-up of solutions of a quasilinear wave equation with nonlinear dissipation. J. Partial Differ. Equ. 15(3), 61–67 (2002)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Benaissa, A.; Mimouni, S.: Energy decay of solutions of a wave equation of \(p\)-Laplacian type with a weakly nonlinear dissipation. J. Inequal. Pure Appl. Math. 7(1), 1–8 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Benaissa, A.; Mimouni, S.: Energy decay of solutions of a wave equation of \(p\)-Laplacian type with a weakly nonlinear dissipation. JIPM. J. Inequal. Pure Appl. Math. 7(1), 8 (2006). Article 15MathSciNetzbMATHGoogle Scholar
  16. 16.
    Benaissa, A.; Messaoudi, S.A.: Exponential decay of solutions of a nonlinearly damped wave equation. NoDEA Nonlinear Differ. Equ. Appl. 12(4), 391–399 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Benaissa, A.; Mokeddem, S.: Decay estimates for the wave equation of \(p\)-Laplacian type with dissipation of \(m\)-Laplacian type. Math. Methods Appl. Sci. 30(2), 237–247 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Benaissa, A.; Guesmia, A.: Global existence and general decay estimates of solutions for degenerate or nondegenerate Kirchhoff equation with general dissipation. Diff. Inte. Equ. 11(1), 1–35 (2011)zbMATHGoogle Scholar
  19. 19.
    Cavalcanti, M.; Guesmia, A.: General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. Differ. Integral Equ. 18(5), 583–600 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Cavalcanti, M.; Cavalcanti, V.D.; Tebou, L.: Stabilization of the wave equation with localized compensating frictional and Kelvin–Voigt dissipating mechanisms. Electron. J. Differ. Equ. 2017(83), 1–18 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Chen, C.; Yao, H.; Shao, L.: Global existence, uniqueness, and asymptotic behavior of solution for \(p\)-Laplacian type wave equation. J. Inequal. Appl. 2010(1), 216760 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ferreira, J.; Messaoudi, S.A.: On the general decay of a nonlinear viscoelastic plate equation with a strong damping and \(\overrightarrow{p}(x, t)\)-Laplacian. Nonlinear Anal. Theory Methods Appl. 104, 40–49 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Galaktionov, V.A.; Pohozaev, S.I.: Blow-up and critical exponents for nonlinear hyperbolic equations. Nonlinear Anal. Theory Methods Appl. 53(3), 453–466 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gao, H.; Ma, T.F.: Global solutions for a nonlinear wave equation with the \(p\)-Laplacian operator. Electron. J. Qual. Theory of Differ. Equ. 11, 1–13 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gao, Y.; Gao, W.: Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents. Bound. Value Probl. 2013(1), 1–8 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Georgiev, V.; Todorova, G.: Existence of solutions of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 109(2), 295–308 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Guesmia, A.: On the decay estimates for elasticity systems with some localized dissipations. Asymptot. Anal. 22(1), 1–13 (2000)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Guesmia, A.: A new approach of stabilization of nondissipative distributed systems. SIAM J. Control Optim. 42(1), 24–52 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Guesmia, A.; Messaoudi, S.A.: Decay estimates of solutions of a nonlinearly damped semilinear wave equation. In: Annales Polonici Mathematici, vol. 85, pp. 25–36. Instytut Matematyczny Polskiej Akademii Nauk (2005)Google Scholar
  30. 30.
    Guo, B.; Gao, W.: Blow-up of solutions to quasilinear hyperbolic equations with \(p(x, t)\)-Laplacian and positive initial energy. C. R. Mec. 342(9), 513–519 (2014)CrossRefGoogle Scholar
  31. 31.
    Hamidi, A.E.; Vetois, J.: Sharp Sobolev asymptotics for critical anisotropic equations. Arch. Ration. Mech. Anal. 192(1), 1–36 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ibrahim, S.; Lyaghfouri, A.: Blow-up solutions of quasilinear hyperbolic equations with critical Sobolev exponent. Math. Modell. Nat. Phenom. 7(2), 66–76 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kafini, M.; Messaoudi, S.A.: A blow-up result in a nonlinear wave equation with delay. Mediterr. J. Math. 13(1), 237–247 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kalantarov, V.K.; Ladyzhenskaya, O.A.: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type. J. Soviet Math. 10(1), 53–70 (1978)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kirchhoff, G.: Vorlesungen uber Mechanik. Teubner, Leipzig (1883)zbMATHGoogle Scholar
  36. 36.
    Komornik, V.; Zuazua, E.: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69, 33–54 (1990)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Komornik, V.: Decay estimates for the wave equation with internal damping. Int. Ser. Numer. Math. 118, 253–266 (1994)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Komornik, V.: Exact Controllability and Stabilization. The Multiplier Method. Masson-John Wiley, Paris (1994)zbMATHGoogle Scholar
  39. 39.
    Korpusov, M.O.: Non-existence of global solutions to generalized dissipative Klein–Gordon equations with positive energy. Electron. J. Differ. Equ. 2012(119), 1–10 (2012)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Kovacik, O.; Rakosnik, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czechoslov. Math. J. 41(4), 592–618 (1991)zbMATHGoogle Scholar
  41. 41.
    Lars, D.; Harjulehto, P.; Hasto, P.; Ruzicka, M.: Lebesgue and Sobolev spaces with variable exponents. In: Lecture Notes in Mathematics, vol. 2017 Springer-Verlag Berlin Heidelberg (2011)Google Scholar
  42. 42.
    Larson, S.; Thome, V.: Partial differential equations with numerical methods. In: Text in Applied Mathematics, vol 45. Springer (2009)Google Scholar
  43. 43.
    Lasiecka, I.: Stabilization of wave and plate-like equation with nonlinear dissipation on the boundary. J. Differ. Equ. 79, 340–381 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lasiecka, I.; Tataru, D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507–533 (1993)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Levine, H.A.: Instability and nonexistence of global solutions of nonlinear wave equation of the form \(Pu=Au+F(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Levine, H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equation. SIAM J. Math. Anal. 5(1), 138–146 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Levine, H.A.; Pucci, P.; Serrin, J.: Some remarks on global nonexistence for nonautonomous abstract evolution equations. Contemp. Math. 208, 253–263 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Levine, H.A.; Serrin, J.: Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch. Ration. Mech. Anal. 137(4), 341–361 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Levine, H.A.; Park, S.R.: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl. 228(1), 181–205 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lions, J.L.: Quelques Methodes de Resolution des Problemes Aux Limites Nonlineaires, 2nd edn. Dunod, Paris (2002)Google Scholar
  51. 51.
    Liu, W.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73(6), 1890–1904 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Lacroix-Sonrier, M.-T.: Distrubutions, Espace de Sobolev, Applications. Ellipses (1998)Google Scholar
  53. 53.
    Martinez, P.: A new method to decay rate estimates for dissipative systems. ESIM Control Optim. Cal. Var. 4, 419–444 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Martinez, P.: A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12(1), 251–283 (1999)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Messaoudi, S.A.: Blow up in a nonlinearly damped wave equation. Math. Nachr. 231(1), 1–7 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Messaoudi, S.A.: Decay of the solution energy for a nonlinearly damped wave equation. Arab. J. Sci. Eng. 26(1), 63–68 (2001). Part AMathSciNetGoogle Scholar
  57. 57.
    Messaoudi, S.A.: Blow up in the Cauchy problem for a nonlinearly damped wave equation. Commun. Appl. Anal. 7(3), 379–386 (2003)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Messaoudi, S.A.; Houari, B.S.: Global non-existence of solutions of a class of wave equations with non-linear damping and source terms. Math. Methods Appl. Sci. 27(14), 1687–1696 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Messaoudi, S.A.: On the decay of solutions for a class of quasilinear hyperbolic equations with non-linear damping and source terms. Math. Methods Appl. Sci. 28(15), 1819–1828 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Messaoudi, S.A.; Soufyane, A.: General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal. Real World Appl. 11(4), 2896–2904 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Messaoudi, S.A.; Talahmeh, A.A.: A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities. Appl. Anal. 96(9), 1509–1515 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Messaoudi, S.A.; Talahmeh, A.A.: A blow-up result for a quasilinear wave equation with variable-exponent nonlinearities. Math. Methods Appl. Sci. 1–11 (2017).  https://doi.org/10.1002/mma.4505
  63. 63.
    Messaoudi, S.A.; Talahmeh, A.A.; Al-Smail, J.H.: Nonlinear damped wave equation: existence and blow-up. Comput. Math. Appl.  https://doi.org/10.1016/j.camwa.2017.07.048
  64. 64.
    Mokeddem, S.; Mansour, K.B.W.: The rate at which the energy of solutions for a class of-Laplacian wave equation decays. Int. J. Differ. Equ. 2015, 721503-1–721503-5 (2015)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Mustafa, M.I.; Messaoudi, S.A.: General energy decay rates for a weakly damped wave equation. Commun. Math. Anal. 9(2), 67–76 (2010)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Nakano, H.: Modulared Semi-ordered Linear Spaces. Maruzen Co., Ltd, Tokyo (1950)zbMATHGoogle Scholar
  67. 67.
    Nakano, H.: Topology and Topological Linear Spaces. Maruzen Co., Ltd, Tokyo (1951)Google Scholar
  68. 68.
    Nakao, M.: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60, 542–549 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Nakao, M.: A difference inequality and its applications to nonlinear evolution equations. J. Math. Soc. Jpn. 30, 747–762 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Nakao, M.: Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations. Math Z. 206, 265–275 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Nakao, M.; Ono, K.: Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation. Funkc. Ekvacioj 38, 417–431 (1995)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Nakao, M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305(3), 403–417 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Nishihara, K.; Yamada, Y.: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkc. Ekvacioj 33(1), 151–159 (1990)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Ono, K.: On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type. Math. Methods Appl. Sci. 20, 151–177 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Ono, K.: On global solutions and blow-up solutions of nonlinear Kirchhoff string with nonlinear dissipation. J. Math. Anal. Appl. 216, 321–342 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Ono, K.: Global solvability for degenerate Kirchhoff equations with weak dissipation. Math. Jpn. 50(3), 409–413 (1999)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Orlicz, W.: Uber konjugierte Exponentenfolgen. Stud. Math. 3(1), 200–212 (1931)CrossRefzbMATHGoogle Scholar
  78. 78.
    Pucci, P.; Serrin, J.: Asymptotic stability for nonautonomous dissipative wave system. Commun. Pure Appl. Math. 49, 177–216 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Quarteroni, A.; Sacco, R.; Saleri, F.: Numerical Mathematics. Springer, New-York, 2000, TAM Series n., vol. 37, 2 edn (2007)Google Scholar
  80. 80.
    Quarteroni, A.; Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Heidelberg (1994). SCM Series n. 23 zbMATHGoogle Scholar
  81. 81.
    Rammaha, M.A.; Toundykov, D.: Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources. J. Math. Phys. 56(8), 081503 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Sun, L.; Ren, Y.; Gao, W.: Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources. Comput. Math. Appl. 71(1), 267–277 (2016)MathSciNetCrossRefGoogle Scholar
  83. 83.
    Taniguchi, T.: Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Commun. Pure Appl. Anal. 16(5), 1571–1585 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Todorova, G.: Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms. J. Math. Anal. Appl. 239(2), 213–226 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Todorova, G.; Vitillaro, E.: Blow-up for nonlinear dissipative wave equations in \(\mathbb{R}^n\). J. Math. Anal. Appl. 303(1), 242–257 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    Tsenov, I.V.: Generalization of the problem of best approximation of a function in the space \(L^s\), (Russian) Uch. Zap. Dagest. Gos. Univ. 7, 25–37 (1961)Google Scholar
  87. 87.
    Vitillaro, E.: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149(2), 155–182 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Vitillaro, E.: Global existence for the wave equation with nonlinear boundary damping and source terms. J. Differ. Equ. 186(1), 259–298 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  89. 89.
    Wang, Y.: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett. 22(9), 1394–1400 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Wu, S.T.: Blow-up of solutions for an integro-differential equation with a nonlinear source. Electron. J. Differ. Equ. 2006(45), 1–9 (2006)MathSciNetGoogle Scholar
  91. 91.
    Wu, Y.; Xue, X.: Decay rate estimates for a class of quasilinear hyperbolic equations with damping terms involving \(p\)-Laplacian. J. Math. Phys. 55(12), 121504 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    Yang, Z.: Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms. Math. Methods Appl. Sci. 25(10), 795–814 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  93. 93.
    Ye, Y.: Global nonexistence of solutions for systems of quasilinear hyperbolic equations with damping and source terms. Bound. Value Probl. 2014(1), 251 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  94. 94.
    Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR Izv. 29(1), 33–66 (1987)CrossRefzbMATHGoogle Scholar
  95. 95.
    Zuazua, E.: Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. P.D.E no. 15, 205–235 (1990)Google Scholar
  96. 96.
    Zuazua, E.: Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 70(4), 513–529 (1991)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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