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Israel Journal of Mathematics

, Volume 219, Issue 1, pp 189–213 | Cite as

Exponential stability for the wave equation with degenerate nonlocal weak damping

  • Marcelo M. Cavalcanti
  • Valeria N. Domingos Cavalcanti
  • Marcio A. Jorge Silva
  • Claudete M. Webler
Article

Abstract

A damped nonlinear wave equation with a degenerate and nonlocal damping term is considered. Well-posedness results are discussed, as well as the exponential stability of the solutions. The degeneracy of the damping term is the novelty of this stability approach.

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References

  1. [1]
    K. Agre and M. A. Rammaha, Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions, Differential and Integral Equations 14 (2001), 1315–1331.MathSciNetMATHGoogle Scholar
  2. [2]
    F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Applied Mathematics and Optimization 51 (2005), 61–105.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calculus of Variations and Partial Differential Equations 34 (2009), 377–411.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha and D. Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete and Continuous Dynamical Systems. Series S 2 (2009), 583–608.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    G. Autuori and P. Pucci, Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Variables and Elliptic equations 56 (2011), 715–753.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    G. Autuori and P. Pucci, Local asymptotic stability for polyharmonic Kirchhoff systems, Applicable Analysis 90 (2011), 493–514.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, Journal of Mathematical Analysis and Applications 352 (2009), 149–165.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for nonlinear Kirchhoff systems, Nonlinear Analysis: Real World Applications 10 (2009), 889–909.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quarterly Journal of Mathematics. Oxford 28 (1977), 473–486.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    V. Barbu, I. Lasiecka and M. A. Rammaha, Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, Indiana University Mathematics Journal 56 (2007), 995–1021.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    V. Barbu, I. Lasiecka and M. A. Rammaha, Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms, Control and Cybernetics 34 (2005), 665–687.MathSciNetMATHGoogle Scholar
  12. [12]
    V. Barbu, I. Lasiecka and M. A. Rammaha, Nonlinear wave equations with degenerate damping and source terms, in Control and Boundary Analysis, Lecture Notes in Pure and Applied Mathematics, Vol. 240, Chapman and Hall/CRC, Boca Raton, FL, 2005, pp. 53–62.MATHGoogle Scholar
  13. [13]
    V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Transactions of the American Mathematical Society 357 (2005), 2571–2611.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization 30 (1992), 1024–1065.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Wellposedness and optimal decay rates for wave equation with nonlinear boundary damping-source interaction, Journal of Differential Equations 236 (2007), 407–459.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, Journal of Mathematical Analysis and Applications 291 (2004), 109–127.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    M. M. Cavalcanti, V. N. Domingos Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler–Bernoulli equation with a nonlocal dissipation in general domains, Differential and Integral Equations 17 (2004), 495–510.MathSciNetMATHGoogle Scholar
  18. [18]
    M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Archive for Rational Mechanics and Analysis 197 (2010), 925–964.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result, Transactions of the American Mathematical Society 361 (2009), 4561–4580.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and D. Toundykov, Unified approach to stabilization of waves on compact surfaces by simultaneous interior and boundary feedbacks of unrestricted growth, Applied Mathematics and Optimization 69 (2014), 83–122.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    C. M. Dafermos, Applications of the invariance principle for compact processes. I. Asymptotically dynamical systems, Journal of Differential Equations 9 (1971), 291–299.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Mathematische Zeitschrift 254 (2006), 729–749.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Annales Scientifiques de l’École Normale Supérieure 36 (2003), 525–551.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, Journal of Differential Equations 109 (1994), 295–308.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    R. T. Glassey, Blow-up theorems for nonlinear wave equations, Mathematische Zeitschrift 132 (1973), 183–203.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    A. Haraux, Nonlinear Evolution Equations-Global Behaviour of Solutions, Lecture Notes in Mathematics, Vol. 841, Springer Verlag, Berlin–New York, 1981.CrossRefGoogle Scholar
  27. [27]
    R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential, Journal of the Mathematical Society of Japan 65 (2013), 183–236.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential and Integral Equations 10 (1997), 1075–1092.MathSciNetMATHGoogle Scholar
  29. [29]
    I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differential and Integral Equations 6 (1993), 507–533.MathSciNetMATHGoogle Scholar
  30. [30]
    H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form Pu tt = Au+F(u), Transactions of the American Mathematical Society 192 (1974), 1–21.MathSciNetGoogle Scholar
  31. [31]
    H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Archive for Rational Mechanics and Analysis 137 (1997), 341–361.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    H. A. Levine, S. R. Park and J. M. Serrin, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, Journal of Mathematical Analysis and Applications 228 (1998), 181–205.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    J. L. Lions and W. A. Strauss, Some non-linear evolution equations, Bulletin fe la Société Mathématique de France 93 (1965), 43–96.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    J. J. Lions, Contrôlabilité Exacte Pertubations et Stabilisation de Systèmes Distribués, Recherches en Mathématiques Appliquées, Vols. 8–9, Masson, Paris, 1988.MATHGoogle Scholar
  35. [35]
    P. Marcati, Decay and stability for nonlinear hyperbolic equations, Journal of Differential Equations 55 (1984), 30–58.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptotic Analysis 19 (1999), 1–17.MathSciNetMATHGoogle Scholar
  37. [37]
    P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Revista Matemática Complutense 12 (1999), 251–283.MathSciNetMATHGoogle Scholar
  38. [38]
    P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Israel Journal of Mathematics 119 (2000), 291–324.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation, Israel Journal of Mathematics 95 (1996), 25–42.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    M. Nakao, Decay and global existence for nonlinear wave equations with localized dissipations in general exterior domains, in New Trends in the Theory of Hyperbolic Equations, Operator Theory: Advances and Applications, Vol. 159, Birkhäuser, Basel, 2005, pp. 213–299.CrossRefGoogle Scholar
  41. [41]
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983.Google Scholar
  42. [42]
    D. R. Pitts and M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations, Indiana University Mathematics Journal 51 (2002), 1479–1509.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    P. Pucci and J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems, Communications on Pure and Applied Mathematics 49 (1996), 177–216.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, Journal of Differential Equations 150 (1998), 203–214.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    P. Radu, G. Todorova and B. Yordanov, Decay estimates for wave equations with variable coefficients, Transactions of the American Mathematical Society 362 (2010), 2279–2299.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete and Continuous Dynamical Systems. Series S 2 (2009), 609–629.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Transactions of the American Mathematical Society 354 (2002), 3621–3637.MathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    J. Serrin, G. Todorova and E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential and Integral Equations 16 (2003), 13–50.MathSciNetMATHGoogle Scholar
  49. [49]
    G. Todorova, Cauchy problem for nonlinear wave equations with nonlinear damping and source terms, Comptes Rendus de l’Académie des Sciences. Paris. Série I. Mathématique 326 (1998), 191–196.MathSciNetMATHGoogle Scholar
  50. [50]
    G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Analysis 41 (2000), 891–905.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions, Nonlinear Analysis 67 (2007), 514–544.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Mathematica Japonicea 17 (1972), 173–193.MathSciNetMATHGoogle Scholar
  53. [53]
    G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canadian Journal of Mathematics 32 (1980), 631–643.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  • Marcelo M. Cavalcanti
    • 1
  • Valeria N. Domingos Cavalcanti
    • 1
  • Marcio A. Jorge Silva
    • 2
  • Claudete M. Webler
    • 1
  1. 1.Department of MathematicsState University of MaringáMaringá, PRBrazil
  2. 2.Department of MathematicsState University of LondrinaLondrina, PRBrazil

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