Skip to main content
SpringerLink
Log in

On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

Let Ω be a 𝒞2-bounded domain of ℝd, d = 2,3, and fix Q = (0,T)× Ω with T ∈ (0,+∞). In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂αt+Au=fb(u) in Q where 1<α<2, ∂αt corresponds to the Caputo fractional derivative of order α, 𝒜 is an elliptic operator and the nonlinearity fb ∈ 𝒞1 (ℝ) satisfies fb(0) = 0 and |f″b(u)| ≤ C |u|b-1 for some b > 1. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂αt +Au=f(t,x) in Q. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Beckers and M. Yamamoto, Regularity and uniqueness of solution to linear diffusion equation with multiple time-fractional derivatives. International Series of Numerical Mathematics 164 (2013), 45–55.

    Article  Google Scholar 

  2. N. Burq and G. Lebeau and F. Planchon, Global existence for energy critical waves in 3-D domains. J. Amer. Math. Soc. 21, No 3 (2008), 831–845.

    Article  MathSciNet  Google Scholar 

  3. K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations. Math. Control and Related Fields 6, No 2 (2016), 251–269.

    Article  MathSciNet  Google Scholar 

  4. J. Ginibre and G. Velo, The global Cauchy problem for nonlinear Klein-Gordon equation. Math. Z 189 (1985), 487–505.

    Article  MathSciNet  Google Scholar 

  5. J. Ginibre and G. Velo, Regularity of solutions of critical and subcritical nonlinear wave equations. Nonlinear Anal 22 (1994), 1–19.

    Article  MathSciNet  Google Scholar 

  6. M.G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical non-linearity. Ann. of Math. 132 (1990), 485–509.

    Article  MathSciNet  Google Scholar 

  7. P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman London, 1985.

    MATH  Google Scholar 

  8. S. Ibrahim and R. Jrad, Strichartz type estimates and the wellposedness of an energy critical 2D wave equation in a bounded domain. J. Differ. Equ. 250, No 9 (2011), 3740–3771.

    Article  Google Scholar 

  9. L. Kapitanski, Cauchy problem for a semilinear wave equation, II. Journal of Soviet Mathematics 62 (1992), 2746–2777.

    Article  MathSciNet  Google Scholar 

  10. L. Kapitanski, Weak and yet weaker solutions of semi-linear wave equations. Comm. Partial Differential Equation 19 (1994), 1629–1676.

    Article  MathSciNet  Google Scholar 

  11. Y. Kian, Cauchy problem for semilinear wave equation with timedependent metrics. Nonlinear Anal. 73 (2010), 2204–2212.

    Article  MathSciNet  Google Scholar 

  12. A.A. Kilbas and H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier Amsterdam, 2006.

    MATH  Google Scholar 

  13. M. Keel and T. Tao, Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), 955–980.

    Article  MathSciNet  Google Scholar 

  14. Z. Li and O.Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations. Inverse Problems 32, No 1 (2016), Article # 015004

    Google Scholar 

  15. J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications I. Springer-Verlag Berlin, (1972).

  16. Y. Luchko, Some uniqueness and existence results for the initialboundary-value problems for the generalized time-fractional diffusion equation. Computers and Mathematics with Applications 59 (2010), 1766–1772.

    Article  MathSciNet  Google Scholar 

  17. Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundaryvalue problems. Fract. Calc. Appl. Anal. 19, No 3 (2016), 676–695 10.1515/fca-2016-0036 https://www.degruyter.com/view/j/fca.2016.19.issue-3/issue-files/fca.2016.19.issue-3.xml

    Article  MathSciNet  Google Scholar 

  18. F. Mainardi, On the initial value problem for the fractional diffusionwave equation In: S. Rionero and T. Ruggeri, Waves and Stability in Continuous Media World Scientific Singapore, (1994), 246–251.

    Google Scholar 

  19. C. Miao and B. Zhang, Hs-global well-posedness for semilinear wave equations. J. Math. Anal. Appl 283 (2003), 645–666.

    Article  MathSciNet  Google Scholar 

  20. M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth. Math. Z. 231 (1999), 479–487.

    Article  MathSciNet  Google Scholar 

  21. M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete and Continuous Dynamical Systems 5, No 1 (1999), 215–231.

    Article  MathSciNet  Google Scholar 

  22. I. Podlubny, Fractional Differential Equations Academic Press San Diego, 1999.

    MATH  Google Scholar 

  23. K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382 (2011), 426–447.

    Article  MathSciNet  Google Scholar 

  24. J. Shatah and M. Struwe, Regularity results for nonlinear wave equations. Ann. of Math. 2, No 138 (1993), 503–518.

    Article  MathSciNet  Google Scholar 

  25. J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equation with critical growth. Intern. Math. Research Notices 7 (1994), 303–309.

    Article  MathSciNet  Google Scholar 

  26. I.M. Sokolov and J. Klafter and A. Blumen, Fractional kinetics. Physics Today 55 (2002), 48–54.

    Article  Google Scholar 

  27. R. Strichartz, A priori estimates for the wave equation and some applications. J. Funct. Anal. 5 (1970), 218–235.

    Article  MathSciNet  Google Scholar 

  28. R. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation. Duke Math. J. 44 (1977), 705–714.

    Article  MathSciNet  Google Scholar 

  29. D. Wang and Z. Xia, Pseudo almost solution of semilinear fractional differential equations with Caputo derivatives. Fract. Calc. Appl. Anal. 18, No 4 (2015), 951–971 0.1515/fca-2015-0056 https://www.degruyter.com/view/j/fca.2015.18.issue-4/issue-files/fca.2015.18.issue-4.xml

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aix Marseille Univ., Université de Toulon, CNRS, CPT, Marseille, France

    Yavar Kian

  2. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-9 Komaba Meguro, Tokyo, 153-8914, Japan

    Masahiro Yamamoto

Authors
  1. Yavar Kian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masahiro Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yavar Kian.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kian, Y., Yamamoto, M. On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations. FCAA 20, 117–138 (2017). https://doi.org/10.1515/fca-2017-0006

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2017-0006

MSC 2010

Key Words and Phrases

Search

Navigation