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Global well-posedness for the nonlinear damped wave equation with logarithmic type nonlinearity

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Abstract

The initial boundary value problem for the nonlinear wave equations with damping and logarithmic nonlinearity is investigated in this paper. By making use of modified potential well theory and the technique of Logarithmic-Sobolev inequality, we establish global existence as well as asymptotic behavior of solution, under the assumption that the initial energy is small. Moreover, we obtain an exponential decay which is much faster than the decay in polynomial nonlinear case of Gazzola and Squassina (Ann I H Poincaré AN 23:185–207, 2006). These results generalize and extend work in application of potential well theory to wave equations.

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References

  1. Abbicco M (2015) The threshold of effective damping for semilinear wave equation. Math Methods Appl Sci 38(6):1032–1045

  2. Abbicco M, Lucente S, Reissig M (2015) A shift in the Strauss exponent for semilinear wave equations with a not effective damping. J Differ Equ 259(10):5040–5073

  3. Cavalcanti MM, Domingos Cavalcanti VN, Martinez P (2004) Existence and decay rate estimates for the wave equation with nonlinear doundary damping and source term. J Differ Equ 203:119–158

  4. Esquivel-Avila J (2003) A Characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations. Nonlinear Anal 52:1111–1127

  5. Esquivel-Avila J (2004) Qualitative analysis of a nonlinear wave equation. Discrete Contin Dyn Syst 10:787–804

  6. Fujishima Y (2018) Global existence and blow-up of solutions for the heat equation with exponential nonlinearity. J Differ Equ 264(11):6809–6842

  7. Galstian A (2008) Global existence for the one-dimensional second order semilinear hyperbolic equations. J Math Anal Appl 344(1):76–98

  8. Gazzola F, Squassina M (2006) Global solutions and finite time blow up for damped semilinear wave equations. Ann I H Poincaré AN 23:185–207

  9. Hao J, Wang F (2016) On the decay and blow-up of solution for coupled nonlinear wave equations with nonlinear damping and source terms. Math Methods Appl Sci 40(7):2550–2565

  10. Haraux A (2006) Decay rate of the range component of solutions to some semilinear evolution equations. Nonlinear Differ Equ Appl 13(4):435–445

  11. Hu Y (2018) On the existence of solutions to a one-dimensional degenerate nonlinear wave equation. J Differ Equ 265(1):157–176

  12. Ikeda M, Inui T, Wakasugi Y (2017) The Cauchy problem for the nonlinear damped wave equation with slowly decaying data. Nonlinear Differ Equ Appl 24(2):10–53

  13. Ikehata R, Suzuki T (1996) Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math J 26:475–491

  14. Lai NA, Takamura H (2018) Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case. Nonlinear Anal 168(53):222–237

  15. Lai NA, Takamura H, Wakasa K (2017) Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent. J Differ Equ 263(26):5377–5394

  16. Levine HA, Serrin J (1997) Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch Ration Mech Anal 137(2):341–361

  17. Li QW, Gao WJ, Han YZ (2016) Global existence blow up and extinction for a class of thin-film equation. Nonlinear Anal 147(45):96–109

  18. Lieb EH, Loss M (2001) Analysis. Graduate Studies in Mathematics

  19. Lin J, Nishihara K, Zhai J (2012) Critical exponent for the semilinear wave equation with time-dependent damping. Discrete Contin Dyn Syst 32(12):4307–4320

  20. Lindblad H, Nakamura M, Sogge CD (2013) Remarks on global solutions for nonlinear wave equations under the standard null conditions. J Differ Equ 254(3):1396–1436

  21. Liu YC (2003) On potential wells and vacuum isolating of solutions for semilinear wave equation. J Differ Equ 192(1):155–169

  22. Liu L, Wang M (2006) Global solutions and blow-up of solutions for some hyperbolic systems with damping and source term. Nonlinear Anal 64:69–91

  23. Liu GW, Zhang HW (2014) Blow up at infinity of solutions for integro-differential equation. Appl Math Comput 230:303–314

  24. Liu YC, Zhao JS (2006) On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal 64:2665–2687

  25. Nakao M, Ono K (1993) Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equation. Math Z 214(11):325–342

  26. Otadi M (2019) Simulation and evaluation of second-order fuzzy boundary value problems. Soft Comput 23(20):10463–10475

  27. Palmieri A (2018) Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces. J Differ Equ 461(2):1215–1240

  28. Palmieri A (2019) A global existence result for a semilinear scale-invariant wave equation in even dimension. Math Methods Appl Sci 42(8):2680–2706

  29. Payne LE, Sattinger DH (1975) Saddle points and instability of nonlinear hyperbolic equations. Israel Math J 22:273–303

  30. Said-Houari B (2011) 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(1):115–133

  31. Sattinger DH (1968) On global solution of nonlinear hyperbolic equations. Arch Ration Mech Anal 30:148–172

  32. Seifi A, Lotfi T, Allahviranloo T (2019) A new efficient method using Fibonacci polynomials for solving of first-order fuzzy Fredholm-Volterra integro-differential equations. Soft Comput 23(19):9777–9791

  33. Soga H (2019) Generalization of the Maxwell equation and relation to the elastic equation. Math Methods Appl Sci 42(11):3950–3966

  34. Tsutsumi M (1972) On solutions of semilinear differential equations in a Hilbert space. Math Jpn 17:173–193

  35. Tsutsumi M (1972) Existence and nonexistence of global solutions for nonlinear equations. Publ Res Inst Math 8:211–229

  36. Wakasugi Y (2014) Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete Contin Dyn Syst 34(9):3831–3846

  37. Wakasugi Y (2017) Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients. J Math Anal Appl 447(1):452–487

  38. Wang Y (2014) Finite time blow-up and global solutions for fourth order damped wave equations. J Math Anal Appl 418(2):713–733

  39. Xu RZ, Wang XC, Yang YB (2018) Global solutions and finite time blow-up for fourth order nonlinear damped wave equation. J Math Phys 59(6):1503–1522

  40. Xu R, Wang X, Yang Y (2018) Global solutions and finite time blow-up for fourth order nonlinear damped wave equation. J Math Phys 59(6):1503–1521

  41. Yao K (2015) Uncertain differential equation with jumps. Soft Comput 19(7):2063–2069

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Correspondence to Lu Yang.

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Supported in part by China NSF Grant No. 11601404, Yanta Scholar Fund of Xi’an University of Finance and Economics.

Communicated by Y. Ni.

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Yang, L., Gao, W. Global well-posedness for the nonlinear damped wave equation with logarithmic type nonlinearity. Soft Comput (2020) doi:10.1007/s00500-019-04660-6

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Keywords

  • Damped nonlinear wave equations
  • Logarithmic nonlinearity
  • Global existence
  • Asymptotic behavior