Skip to main content
Top

2024 | OriginalPaper | Chapter

4. Well-Posedness and Regularity of Fractional Wave Equations

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter, we first study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solutions. We also establish a well-posedness result for nonlinear problem. As an application, we discuss a case of time fractional telegraph equations. Section 4.2 studies the semilinear time fractional wave equation on a whole Euclidean space, also known as the superdiffusive equations. Based on the initial data taken in the fractional Sobolev spaces and some known Sobolev embeddings, we prove the local/global well-posedness results of \(L^2\)-solutions for the linear and semilinear problems. In Sect. 4.3, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, and we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension \(1\leq d\leq 3\). Our proofs base on the analyticity of the Mittag-Leffler functions, the framework of prior estimates, and the type of exponential nonlinearity. The material in Sect. 4.1 is due to Zhou and He (Monatsh Math 194(2):425–458, 2021) . The results in Sect. 4.2 are taken from Zhou, He, Alsaedi, and Ahmad Zhou et al. (Elec Res Arch 30(8):2981–3003, 2022). The results in Sect. 4.3 are adopted from He and Zhou (Bull Sci Math 189:103357, 2023).

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
  1. E. Affili, E. Valdinoci, Decay estimates for evolution equations with classical and fractional time-derivatives. J. Diff. Equ. 266(7), 4027–4060 (2019)MathSciNetView Article
  2. M.R. Alaimia, N.E. Tatar, Blow up for the wave equation with a fractional damping. J. Appl. Anal. 11(1), 133–144 (2005)MathSciNetView Article
  3. L. Aloui, S. Ibrahim, M. Khenissi, Energy decay for linear dissipative wave equations in exterior domains. J. Diff. Equ. 259(5), 2061–2079 (2015)MathSciNetView Article
  4. E. Alvarez, C.G. Gal, V. Keyantuo, M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations. Nonlinear Anal. 181, 24–61 (2019)MathSciNetView Article
  5. T.M. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional Calculus with Applications in Mechanics (Wiley-ISTE, New York, 2014)View Article
  6. A. Bekkai, B. Rebiai, M. Kirane, On local existence and blowup of solutions for a time-space fractional diffusion equation with exponential nonlinearity. Math. Methods Appl. Sci. 42, 1819–1830 (2019)MathSciNetView Article
  7. J. Bergh, J. Löfström, Interpolation Spaces: An Introduction (Springer, Berlin, 1976)View Article
  8. T. Cazenave, Semilinear Schrödinger Equations (American Mathematical Society, New York, 2003)View Article
  9. T. Cazenave, B. Weissler, The cauchy problem for the critical nonlinear Schrödinger equation in \(H^s\). Nonlinear Anal. 14(10), 807–836 (1990)
  10. L. Chen, Nonlinear stochastic time-fractional diffusion equations on \(\mathbb {R}\): moments, Hölder regularity and intermittency. Trans. Amer. Math. Soc. 369(12), 8497–8535 (2017)
  11. J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338, 1364–1377 (2008)MathSciNetView Article
  12. M.F. de Alemida, J.C.P. Precioso, Existence and symmetries of solutions in Besov-Morrey spaces for a semilinear heat-wave type equation. J. Math. Anal. Appl. 432, 338–355 (2015)MathSciNetView Article
  13. M.F. de Alemida, A. Viana, Self-similar solutions for a superdiffusive heat equation with gradient nonlinearity. Electr. J. Diff. Equ. 2016(250), 1–20 (2016)MathSciNet
  14. J.D. Djida, A. Fernandez, I. Area, Well-posedness results for fractional semi-linear wave equations. Discrete Contin. Dyn. Syst. Ser. B 25(2), 569–597 (2020)MathSciNet
  15. H. Dong, Y. Liu, Weighted mixed norm estimates for fractional wave equations with VMO coefficients. J. Differ. Equ. 337, 168–254 (2022)MathSciNetView Article
  16. L.C. Evans, Partial Differential Equations, 2nd edn. (American Mathematical Society, Providence, 2010)
  17. A.Z. Fino, M. Kirane, The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Commun. Pure Appl. Anal. 19(7), 3625–3650 (2020)MathSciNetView Article
  18. G. Fragnelli, D. Mugnai, Stability of solutions for some classes of nonlinear damped wave equations. SIAM J. Control Optim. 47(5), 2520–2539 (2008)MathSciNetView Article
  19. Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, 309–321 (1990)MathSciNet
  20. G. Furioli, T. Kawakami, B. Ruf, E. Terraneo, Asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity. J. Differ. Equ. 262, 145–180 (2017)MathSciNetView Article
  21. T. Ghoula, V.T. Nguyen, H. Zaag, Blowup solutions for a reaction-diffusion system with exponential nonlinearities. J. Differ. Equ. 264, 7523–7579 (2018)MathSciNetView Article
  22. I. Graham, U. Langer, J. Melenk, M. Sini (Eds.), Direct and Inverse Problems in Wave Propagation and Applications (Walter de Gruyter, Berlin, 2013)
  23. B. Han, K. Kim, D. Park, Weighted \(L_q(L_p)\)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives. J. Differ. Equ. 269, 3515–3550 (2020)
  24. H. Hirata, C. Miao, Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative. Adv. Differ. Equ. 7(2), 217–236 (2002)MathSciNet
  25. S. Ibrahim, M. Majdoub, N. Masmoudi, K. Nakanishi, Scattering for the two-dimensional energy-critical wave equation. Duke Math. J. 150(2), 287–329 (2009)MathSciNetView Article
  26. R. Ikehata, G. Todorova, B. Yordanov, Wave equations with strong damping in Hilbert spaces. J.Differ. Equ. 254(8), 3352–3368 (2013)MathSciNetView Article
  27. N. Ioku, The Cauchy problem for heat equations with exponential nonlinearity. J. Differ. Equ. 251, 1172–1194 (2011)MathSciNetView Article
  28. N. Ioku, B. Ruf, E. Terraneo, Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in \({\mathbb R}^2\). Math. Phys. Anal. Geom. 18(29), 19 (2015)
  29. Y. Kian, M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations. Fract. Calc. Appl. Anal. 20(1), 117–138 (2017)MathSciNetView Article
  30. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier Science B.V., Amsterdam, 2006)
  31. I. Kim, K.H. Kim, S. Lim, An \(L_q(L_ p)\)-theory for the time fractional evolution equations with variable coefficients. Adv. Math. 306, 123–176 (2017)
  32. H. Kita, On interpolation of the Fourier maximal operator in Orlicz spaces. Acta Math. Hungar. 81(3), 175–193 (1998)MathSciNetView Article
  33. L. Li, J.G. Liu, L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation. J. Differ. Equ. 265(3), 1044–1096 (2018)MathSciNetView Article
  34. C. Lin, G. Nakamura, Unique continuation property for anomalous slow diffusion equation. Comm. Partial Differ. Equ. 41(5), 749–758 (2016)MathSciNetView Article
  35. C. Lin, G. Nakamura, Unique continuation property for multi-terms time fractional diffusion equations. Math. Ann. 373(3–4), 929–952 (2019)MathSciNetView Article
  36. Y. Luchko, Fractional wave equation and damped waves. J. Math. Phys. 54(3), 031505 (2013)
  37. Y. Luchko, F. Mainardi, Fractional diffusion-wave phenomena. Handbook Fract. Calculus Appl. 5, 71–98 (2019)MathSciNet
  38. O. Mahouachi, T. Saanouni, Well and ill-posedness issues for a class of 2D wave equation with exponential growth. J. Partial Differ. Equ. 24(4), 361–384 (2011)MathSciNetView Article
  39. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, An Introduction to Mathematical Models (Imperial College Press, London, 2010)View Article
  40. M. Majdoub, S. Tayachi, Well-posedness, Global existence and decay estimates for the heat equation with general power-exponential nonlinearities. Proc. Int. Cong. of Math. Rio de Janeiro 2, 2379–2404 (2018)
  41. M. Majdoub, S. Otsmane, S. Tayachi, Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity. Adv. Differ. Equ. 23, 489–522 (2018)MathSciNet
  42. R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)MathSciNetView Article
  43. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)
  44. E. Orsingher, L.Beghin, Time-fractional telegraph equations and telegraph process with Brownian time. Probab. Theory Relat. Fields 128(1), 141–160 (2004)View Article
  45. E. Otárola, A.J. Salgado, Regularity of solutions to space-time fractional wave equations: a PDE approach. Fract. Calc. Appl. Anal. 21(5), 1262–1293 (2018)MathSciNetView Article
  46. H. Pecher, Local solutions of semilinear wave equations in \(H^{s+1}\). Math. Methods Appl. Sci. 19(2), 145–170 (1996)
  47. I. Podlubny, Fractional Differential Equations (Academic, San Diego, 1999)
  48. T. Saanouni, A blowing up wave equation with exponential type nonlinearity and arbitrary positive energy. J. Math. Anal. Appl. 421, 444–452 (2015)MathSciNetView Article
  49. K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011)MathSciNetView Article
  50. W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30(1), 134–144 (1989)MathSciNetView Article
  51. M. Struwe, Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions. Math. Ann. 350, 707–719 (2011)MathSciNetView Article
  52. M. Stojanovic, R. Gorenflo, Nonlinear two term time fractional diffusion wave problem. Nonlinear Anal. Real World Appl. 11(5), 3512–3523 (2010)MathSciNetView Article
  53. M. Suzuki, Local existence and nonexistence for reaction-diffusion systems with coupled exponential nonlinearities. J. Math. Anal. Appl. 477(1), 776–804 (2019)MathSciNetView Article
  54. N.E. Tatar, A blow up result for a fractionally damped wave equation. NoDEA Nonlinear Differ. Equ. Appl. 12(2), 215–226 (2005)MathSciNetView Article
  55. H. Vivian, J. Pym, M. Cloud, Applications of Functional Analysis and Operator Theory (Elsevier, Amsterdam, 2005)
  56. H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)MathSciNetView Article
  57. R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations. Math. Ann. 356, 99–146 (2013)MathSciNetView Article
  58. E. Zeidler, Nonlinear Functional Analysis and Its Application II/A (Springer, Berlin, 1990)
  59. Q. Zhang, Y. Li, Global well-posedness and blow-up solutions of the Cauchy problem for a time-fractional superdiffusion equation. J. Evol. Equ. 19, 271–303 (2019)MathSciNetView Article
  60. Y. Zhou, Basic Theory of Fractional Differential Equations (World Scientific, Singapore, 2014)View Article
  61. Y. Zhou, J.W. He, Well-posedness and regularity for fractional damped wave equations. Monatsh. Math. 194(2), 425–458 (2021)MathSciNetView Article
Metadata
Title
Well-Posedness and Regularity of Fractional Wave Equations
Author
Yong Zhou
Copyright Year
2024
DOI
https://doi.org/10.1007/978-3-031-74031-2_4

Premium Partner