Osaka Journal of Mathematics

Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms

Ryo Ikehata and Hiroshi Takeda

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Abstract

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$. As is pointed out by [10], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Especially, the result obtained for the $n = 3$ case is essentially new. Our proof is based on several estimates for the corresponding linear equations.

Article information

Source
Osaka J. Math., Volume 56, Number 4 (2019), 807-830.

Dates
First available in Project Euclid: 21 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1571623223

Mathematical Reviews number (MathSciNet)
MR4020638

Zentralblatt MATH identifier
07144186

Subjects
Primary: 35L15: Initial value problems for second-order hyperbolic equations 35L05: Wave equation
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Ikehata, Ryo; Takeda, Hiroshi. Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms. Osaka J. Math. 56 (2019), no. 4, 807--830. https://projecteuclid.org/euclid.ojm/1571623223


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References

  • P. Brenner, V. Thomée and L. Wahlbin: Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, 434. Springer-Verlag, Berlin-New York, 1975.
  • M. D'Abbicco and M. Reissig: Semilinear structural damped waves, Math. Methods Appl. Sci. 37 (2014), 1570–1592.
  • M. Giga, Y. Giga and J. Saal: Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions, Progress in Nonlinear Differential Equations and their Applications, 79. Birkhäuser Boston, Inc., Boston, MA, 2010.
  • N. Hayashi, E.I. Kaikina and P.I. Naumkin: Damped wave equation with super critical nonlinearities, Differential Integral Equations 17 (2004), 637–652.
  • T. Hosono and T. Ogawa: Large time behavior and $L^{p}$-$L^{q}$ estimate of $2$-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004), 82–118.
  • D. Hoff and K. Zumbrun: Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 (1995), 603–676.
  • R. Ikehata: Asymptotic profiles for wave equations with strong damping, J. Differential Equations 257 (2014), 2159–2177.
  • R. Ikehata, Y. Miyaoka and T. Nakatake: Decay estimates of solutions for dissipative wave equations in $\textbf{R}^N$ with lower power nonlinearities, J. Math. Soc. Japan 56 (2004), 365–373.
  • R. Ikehata and M. Onodera: Remarks on large time behavior of the L2-norm of solutions to strongly damped wave equations, Differential Integral Equations 30 (2017), 505–520.
  • R. Ikehata and A. Sawada: Asymptotic profiles of solutions for wave equations with frictional and viscoelastic damping terms, Asymptotic Anal. 98 (2016), 59–77.
  • R. Ikehata and H. Takeda: Critical exponent for nonlinear wave equations with frictional and viscoelastic damping terms, Nonlinear Anal. 148 (2017), 228–253.
  • R. Ikehata and K. Tanizawa: Global existence of solutions for semilinear damped wave equations in $\textbf{R}^{N}$ with noncompactly supported initial data, Nonlinear Anal. 61 (2005), 1189–1208.
  • R. Ikehata, G. Todorova and B. Yordanov: Wave equations with strong damping in Hilbert spaces, J. Differential Equations 254 (2013), 3352–3368.
  • Y. Kagei and T. Kobayashi: Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal. 177 (2005), 231–330.
  • G. Karch: Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math. 143 (2000), 175–197.
  • T. Kawakami and Y. Ueda: Asymptotic profiles to the solutions for a nonlinear damped wave equation, Differential Integral Equations 26 (2013), 781–814.
  • T. Kobayashi and Y. Shibata: Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations, Pacific J. Math. 207 (2002), 199–234.
  • P. Marcati and K. Nishihara: The $L^{p}$-$L^{q}$ estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003), 445–469.
  • T. Narazaki: $L^{p}$-$L^{q}$ estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004), 585–626.
  • K. Nishihara: $L^{p}$-$L^{q}$ estimates to the damped wave equation in $3$-dimensional space and their application, Math. Z. 244 (2003), 631–649.
  • G. Ponce: Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985), 399–418.
  • I. Segal: Dispersion for non-linear relativistic equations. II, Ann. Sci. École Norm. Sup. (4) 1 (1968), 459–497.
  • Y. Shibata: On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci. 23 (2000), 203–226.
  • H. Takeda: Higher-order expansion of solutions for a damped wave equation, Asymptot. Anal. 94 (2015), 1–31.
  • G. Todorova and B. Yordanov: Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001), 464–489.
  • Q.S. Zhang: A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 109–114.