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Exponential stability for the wave equation with degenerate nonlocal weak damping

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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. 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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Autuori and P. Pucci, Local asymptotic stability for polyharmonic Kirchhoff systems, Applicable Analysis 90 (2011), 493–514.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for nonlinear Kirchhoff systems, Nonlinear Analysis: Real World Applications 10 (2009), 889–909.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quarterly Journal of Mathematics. Oxford 28 (1977), 473–486.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. M. Dafermos, Applications of the invariance principle for compact processes. I. Asymptotically dynamical systems, Journal of Differential Equations 9 (1971), 291–299.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. T. Glassey, Blow-up theorems for nonlinear wave equations, Mathematische Zeitschrift 132 (1973), 183–203.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Haraux, Nonlinear Evolution Equations-Global Behaviour of Solutions, Lecture Notes in Mathematics, Vol. 841, Springer Verlag, Berlin–New York, 1981.

    Book  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential and Integral Equations 10 (1997), 1075–1092.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. L. Lions and W. A. Strauss, Some non-linear evolution equations, Bulletin fe la Société Mathématique de France 93 (1965), 43–96.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. J. Lions, Contrôlabilité Exacte Pertubations et Stabilisation de Systèmes Distribués, Recherches en Mathématiques Appliquées, Vols. 8–9, Masson, Paris, 1988.

    MATH  Google Scholar 

  35. P. Marcati, Decay and stability for nonlinear hyperbolic equations, Journal of Differential Equations 55 (1984), 30–58.

    Article  MathSciNet  MATH  Google Scholar 

  36. P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptotic Analysis 19 (1999), 1–17.

    MathSciNet  MATH  Google Scholar 

  37. P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Revista Matemática Complutense 12 (1999), 251–283.

    MathSciNet  MATH  Google Scholar 

  38. P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Israel Journal of Mathematics 119 (2000), 291–324.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation, Israel Journal of Mathematics 95 (1996), 25–42.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  43. P. Pucci and J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems, Communications on Pure and Applied Mathematics 49 (1996), 177–216.

    Article  MathSciNet  MATH  Google Scholar 

  44. P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, Journal of Differential Equations 150 (1998), 203–214.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  50. G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Analysis 41 (2000), 891–905.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  52. H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Mathematica Japonicea 17 (1972), 173–193.

    MathSciNet  MATH  Google Scholar 

  53. G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canadian Journal of Mathematics 32 (1980), 631–643.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

    Marcelo M. Cavalcanti, Valeria N. Domingos Cavalcanti & Claudete M. Webler

  2. Department of Mathematics, State University of Londrina, 86057-970, Londrina, PR, Brazil

    Marcio A. Jorge Silva

Authors
  1. Marcelo M. Cavalcanti
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  2. Valeria N. Domingos Cavalcanti
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  3. Marcio A. Jorge Silva
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  4. Claudete M. Webler
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Correspondence to Marcelo M. Cavalcanti.

Additional information

Research of Marcelo M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0

Research of Valéria N. Domingos Cavalcanti partially supported by the CNPq Grant 304895/2003-2

Research of Marcio A. Jorge Silva partially supported by the CNPq Grant 441414/2014-1

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Cavalcanti, M.M., Domingos Cavalcanti, V., Jorge Silva, M. et al. Exponential stability for the wave equation with degenerate nonlocal weak damping. Isr. J. Math. 219, 189–213 (2017). https://doi.org/10.1007/s11856-017-1478-y

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  • DOI: https://doi.org/10.1007/s11856-017-1478-y

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