Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ASYMPTOTIC BEHAVIORS OF FUNDAMENTAL SOLUTION AND ITS DERIVATIVES TO FRACTIONAL DIFFUSION-WAVE EQUATIONS

  • Received : 2015.06.09
  • Published : 2016.07.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let p(t, x) be the fundamental solution to the problem $${\partial}^{\alpha}_tu=-(-{\Delta})^{\beta}u,\;{\alpha}{\in}(0,2),\;{\beta}{\in}(0,{\infty})$$. If ${\alpha},{\beta}{\in}(0,1)$, then the kernel p(t, x) becomes the transition density of a Levy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivatives $$D^n_x(-{\Delta}_x)^{\gamma}D^{\sigma}_tI^{\delta}_tp(t,x),\;{\forall}n{\in}{\mathbb{Z}}_+,\;{\gamma}{\in}[0,{\beta}],\;{\sigma},{\delta}{\in}[0,{\infty})$$, where $D^n_x$ x is a partial derivative of order n with respect to x, $(-{\Delta}_x)^{\gamma}$ is a fractional Laplace operator and $D^{\sigma}_t$ and $I^{\delta}_t$ are Riemann-Liouville fractional derivative and integral respectively.

Keywords

Acknowledgement

Supported by : Samsung Science and Technology Foundation

References

  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with formulas, graphs, and mathematical tables, Number 55, Courier Dover Publications, 1972.
  2. G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999.
  3. V. V. Anh and N. N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104 (2001), no. 5-6, 1349-1387. https://doi.org/10.1023/A:1010474332598
  4. B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of barnes-integrals, Compositio Math. 15 (1964), 239-341.
  5. Z.-Q. Chen, K.-H. Kim, and P. Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl. 125 (2015), no. 4, 1470-1499. https://doi.org/10.1016/j.spa.2014.11.005
  6. Z.-Q. Chen, M. M. Meerschaert, and E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl. 393 (2012), no. 2, 479-488. https://doi.org/10.1016/j.jmaa.2012.04.032
  7. P. Clement, G. Gripenberg, and S.-O. Londen, Schauder estimates for equations with fractional derivatives, Trans. Amer. Math. Soc. 352 (2000), no. 5, 2239-2260. https://doi.org/10.1090/S0002-9947-00-02507-1
  8. P. Clement, S.-O. Londen, and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Equations 196 (2004), no 2, 418-447. https://doi.org/10.1016/j.jde.2003.07.014
  9. S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Operator Theory: Advances and Applications, 152. Birkhauser Verlag, Basel, 2004.
  10. S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), no. 2, 211-255. https://doi.org/10.1016/j.jde.2003.12.002
  11. R. Goreno, A. Iskenderov, and Y. Luchko, Mapping between solutions of fractional diffusion-wave equations, Fract. Calc. Appl. Anal. 3 (2000), no. 1, 75-86.
  12. A. Hanyga, Multidimensional solutions of space-time-fractional diffusion equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2020, 933-957. https://doi.org/10.1098/rspa.2001.0904
  13. S. Jo and M. Yang, Precise asymptotic approximations for kernels corresponding to Levy processes, Potential Anal. 40 (2014), no. 3, 203-230. https://doi.org/10.1007/s11118-013-9346-9
  14. A. A. Kilbas, H-transforms: Theory and Applications, CRC Press, 2004.
  15. I. Kim and K.-H. Kim, A generalization of the Littlewood-Paley inequality for the fractional Laplacian $(-{\Delta})^{{\alpha}/2}$, J. Math. Anal. Appl. 388 (2012), no. 1, 175-190. https://doi.org/10.1016/j.jmaa.2011.11.031
  16. I. Kim, K.-H. Kim, and P. Kim, Parabolic Littlewood-Paley inequality for ${\phi}(-{\Delta})$-type operators and applications to stochastic integro-differential equations, Adv. Math. 249 (2013), 161-203. https://doi.org/10.1016/j.aim.2013.09.008
  17. I. Kim, K.-H. Kim, and S. Lim, An $L_q(L_p)$-theory for the time fractional evolution equations with variable coecients, arXiv preprint arXiv:1505.00504, 2015.
  18. I. Kim, K.-H. Kim, and S. Lim, An $L_q(L_p)$-theory for parabolic pseudo-differential equations, Calderon-Zygmund approach, arXiv preprint arXiv:1503.04521, 2015.
  19. I. Kim, K.-H. Kim, and S. Lim, Parabolic BMO estimates for pseudo-differential operators of arbitrary order, J. Math. Anal. Appl. 427 (2015), no. 2, 557-580. https://doi.org/10.1016/j.jmaa.2015.02.065
  20. A. N. Kochubei, Asymptotic properties of solutions of the fractional diffusion-wave equation, Fract. Calc. Appl. Anal. 17 (2014), no. 3, 881-896. https://doi.org/10.2478/s13540-014-0203-3
  21. N. V. Krylov, On the foundation of the $L_p)$-theory of stochastic partial differential equations, Stochastic Partial Differential Equations and Applications-VII, page 179-191, 2006.
  22. Z. Li, Y. Luchko, and M. Yamamoto, Asymptotic estimates of solutions to initial- boundary-value problems for distributed order time-fractional diffusion equations, Fract. Calc. Appl. Anal. 17 (2014), no. 4, 1114-1136. https://doi.org/10.2478/s13540-014-0217-x
  23. M. Magdziarz and R. Schilling, Asymptotic properties of Brownian motion delayed by inverse subordinators, Proc. Amer. Math. Soc. 143 (2015), no. 10, 4485-4501. https://doi.org/10.1090/proc/12588
  24. F. Mainardi, Fractional diffusive waves in viscoelastic solids, Nonlinear Waves in Solids, pages 93-97, Fairfield, 1995.
  25. M. M. Meerschaert, R. L. Schilling, and A. Sikorskii, Stochastic solutions for fractional wave equations, Nonlinear Dynam. 80 (2015), no. 4, 1685-1695. https://doi.org/10.1007/s11071-014-1299-z
  26. R. Metzler, E. Barkai, and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional fokker-planck equation approach, Phys. Rev. Lett. 82 (1999), no. 18, 3563. https://doi.org/10.1103/PhysRevLett.82.3563
  27. R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (2000), no. 1-2, 107-125. https://doi.org/10.1016/S0378-4371(99)00503-8
  28. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 1-77. https://doi.org/10.1016/S0370-1573(00)00070-3
  29. R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, 161-208. https://doi.org/10.1088/0305-4470/37/1/011
  30. I. Podlubny, Fractional Differential Equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, volume 198, Academic press, 1998.
  31. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions, volume 2, CRC Press, 1998.
  32. 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), no. 1, 426-447. https://doi.org/10.1016/j.jmaa.2011.04.058
  33. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, New York, 1993.
  34. W. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), no. 1, 134-144. https://doi.org/10.1063/1.528578
  35. E. M. Stein and G. L. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, volume 1, Princeton university press, 1971.
  36. J. van Neerven, M. Veraar, L. Weis, et al., Stochastic maximal lp-regularity, The Annals of Probability 40 (2012), no. 2, 788-812. https://doi.org/10.1214/10-AOP626
  37. R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ. 5 (2005), no. 1, 79-103. https://doi.org/10.1007/s00028-004-0161-z

Cited by

  1. Representation of solutions and large-time behavior for fully nonlocal diffusion equations vol.263, pp.1, 2017, https://doi.org/10.1016/j.jde.2017.02.030
  2. Some Compactness Criteria for Weak Solutions of Time Fractional PDEs vol.50, pp.4, 2018, https://doi.org/10.1137/17M1145549
  3. A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus pp.1661-8262, 2019, https://doi.org/10.1007/s11785-018-00887-7