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General Stability and Exponential Growth for a Class of Semi-linear Wave Equations with Logarithmic Source and Memory Terms

  • Amir Peyravi
Article

Abstract

In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation
$$\begin{aligned} u_{tt}-\Delta u + u + (g\,*\, \Delta u)(t)+ h(u_{t})u_{t}+|u|^{2}u=u\log |u|^{k}, \end{aligned}$$
in an open bounded domain \(\Omega \subseteq \mathbb {R}^3\) whith \(h(s)=k_{0}+k_{1}|s|^{m-1}.\) We prove a general stability of solutions which improves and extends some previous studies such as the one by Hu et al. (Appl Math Optim,  https://doi.org/10.1007/s00245-017-9423-3) in the case \(g=0\) and in presence of linear frictional damping \(u_{t}\) when the cubic term \(|u|^2u\) is replaced with u. In the case \(k_{1}=0,\) we also prove that the solutions will grow up as an exponential function. Our result shows that the memory kernel g dose not need to satisfy some restrictive conditions to cause the unboundedness of solutions.

Keywords

General decay Exponential growth Logarithmic wave equations Memory damping 

Mathematics Subject Classification

35B35 35B40 74D10 93D20 

References

  1. 1.
    Al-Gharabli, M.M., Messaoudi, S.A.: The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term. J. Math. Anal. Appl. 454, 1114–1128 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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. (2017).  https://doi.org/10.1007/s00028-017-0392-4
  3. 3.
    Barrow, J., Parsons, P.: Inflationary models with logarithmic potentials. Phys. Rev. D 52, 5576–5578 (1995)CrossRefGoogle Scholar
  4. 4.
    Bartkowski, J.D., Górka, P.: One-dimensional Klein-Gordon equation with logarithmic nonlinearities. J. Phys. A 41(35), 355201 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berrimi, S., Messaoudi, S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 2004(88), 1–10 (2004)MathSciNetMATHGoogle Scholar
  6. 6.
    Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 2002(44), 1–14 (2002)MathSciNetMATHGoogle Scholar
  7. 7.
    Cazenave, T., Haraux, A.: Equations devolution avec non-linearite logarithmique. Ann. Fac. Sci. Toulouse Math. 2(1), 21–51 (1980)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, G.W., Wang, Y.P., Wang, S.B.: Initial boundary value problem of the generalized cubic double dispersion equation. J. Math. Anal. Appl. 299, 563–577 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Doddato, F., McDonald, J.: New Q-ball solutions in gauge-mediation. Affleck-Dine baryogenesis and gravitino dark matter. J. Cosmol. Astropart. Phys. 6, 031 (2012)CrossRefGoogle Scholar
  10. 10.
    Enqvist, K., McDonald, J.: Q-balls and baryogenesis in the MSSM. Phys. Lett. B 425, 309–321 (1998)CrossRefGoogle Scholar
  11. 11.
    Georgiev, V., Todorova, G.: Existence of a solution of the wave equation with nonlinear damping and source terms. J. Differ. Equ. 109(2), 295–308 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gerbi, S., Said-Houari, B.: Exponential decay for solutions to semilinear damped wave equation. Discret. Contin. Dyn. Syst. S 5(3), 559–566 (2012)MathSciNetMATHGoogle Scholar
  13. 13.
    Gerbi, S., Said-Houari, B.: Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions. Adv. Nonlinear Anal. 2(2), 163–193 (2013)MathSciNetMATHGoogle Scholar
  14. 14.
    Górka, P.: Logarithmic Klein–Gordon equation. Acta Phys. Polon. B 40(1), 59–66 (2009)MathSciNetMATHGoogle Scholar
  15. 15.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Han, X.: Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics. Bull. Korean Math. Soc. 50(1), 275–283 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hiramatsu, T., Kawasaki, M., Takahashi, F.: Numerical study of Q-ball formation in gravity mediation. J. Cosmol. Astropart. Phys. 6, 008 (2010)CrossRefGoogle Scholar
  18. 18.
    Hu, Q., Zhang, H.: Initial boundary value problem for generalized logarithmic improved Boussinesq equation (2017).  https://doi.org/10.1002/mma.4255
  19. 19.
    Hu, Q., Zhang, H., Liu, G.: Asymptotic behavior for a class of logarithmic wave equations with linear damping. Appl. Math. Optim.  https://doi.org/10.1007/s00245-017-9423-3
  20. 20.
    Hu, Q., Zhang, H., Liu, G.: Global existence and exponential growth of solution for the logarithmic Boussinesq-type equation. J. Math. Anal. Appl. 436, 990–1001 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ikehata, R., Suzuki, T.: Stable and unstable sets for evolution equation for parabolic and hyperbolic type. Hiroshima Math. J. 26, 475–491 (1996)MathSciNetMATHGoogle Scholar
  22. 22.
    Levine, H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form \(Pu_{tt}=Au+F(u)\). Trans. Am. Math. Soc. 192, 1–21 (1974)MATHGoogle Scholar
  23. 23.
    Liu, W.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73, 1890–1904 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Liu, Y.C., Xu, R.Z.: Potential well method for initial boundary value problem of the generalized double dispersion equations. Commun. Pure Appl. Anal. 7, 63–81 (2008)MathSciNetMATHGoogle Scholar
  25. 25.
    Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Messaoudi, S.A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58–66 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Messaoudi, S.A.: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J. Math. Anal. Appl. 320, 902–915 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273–303 (1975)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pişkin, E., Uysal, T., Hajrulla, S.: Exponential decay of solutions for a higher order wave equation with logarithmic source term. Preprints (2018).  https://doi.org/10.20944/preprints201802.0057.v1
  30. 30.
    Said-Houari, B.: Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. Z. Angew. Math. Phys. 62, 115–133 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wang, Y., Wang, Y.: Exponential energy decay of solutions of viscoelastic wave equations. J. Math. Anal. Appl. 347, 18–25 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wazwaz, A.-M.: Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation. Ocean Eng. 94, 111–115 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShiraz UniversityShirazIran

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