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On existence and uniqueness of solutions for semilinear fractional wave equations

  • Yavar Kian EMAIL logo and Masahiro Yamamoto
Published/Copyright: February 19, 2017
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 1

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 ∂tαu+Au=fb(u)inQwhere1<α<2,∂tα 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 |fb′(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 ∂tαu+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.

MSC 2010: Primary 35R11; Secondary 35D30
Key Words and Phrases: well posedness; Strichartz estimates; fractional wave equation; nonlinear equation

Acknowledgements

The second author is partially supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science.

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Received: 2016-7-28
Published Online: 2017-2-19
Published in Print: 2017-2-1

© 2017 Diogenes Co., Sofia

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https://doi.org/10.1515/fca-2017-0006
Keywords for this article
well posedness; Strichartz estimates; fractional wave equation; nonlinear equation

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 20–1–2017)
  4. Survey paper
  5. Ten equivalent definitions of the fractional laplace operator
  6. Research paper
  7. Consensus of fractional-order multi-agent systems with input time delay
  8. Research paper
  9. Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives
  10. Research paper
  11. A preconditioned fast finite difference method for space-time fractional partial differential equations
  12. Research paper
  13. On existence and uniqueness of solutions for semilinear fractional wave equations
  14. Research paper
  15. Computational solutions of the tempered fractional wave-diffusion equation
  16. Research paper
  17. Completeness on the stability criterion of fractional order LTI systems
  18. Research paper
  19. Wavelet convolution product involving fractional fourier transform
  20. Research paper
  21. Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method
  22. Research paper
  23. A foundational approach to the Lie theory for fractional order partial differential equations
  24. Research paper
  25. Null-controllability of a fractional order diffusion equation
  26. Research paper
  27. New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions
  28. Research paper
  29. The stretched exponential behavior and its underlying dynamics. The phenomenological approach
  30. Short Paper
  31. Lyapunov-type inequality for an anti-periodic fractional boundary value problem
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