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Regularity of solutions to space–time fractional wave equations: A PDE approach

Enrique Otárola EMAIL logo and Abner J. Salgado


We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.


EO is partially supported by CONICYT through FONDECYT Project 3160201. AJS is partially supported by NSF Grant DMS-1720213.


[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics (1964).Search in Google Scholar

[2] N.I. Achieser, Theory of Approximation. Dover Publications, Inc., New York (1992).Search in Google Scholar

[3] R.A. Adams Sobolev Spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975).Search in Google Scholar

[4] I. Athanasopoulos and L.A. Caffarelli, Continuity of the temperature in boundary heat control problems. Adv. Math. 224, No 1 (2010), 293–315.10.1016/j.aim.2009.11.010Search in Google Scholar

[5] L. Banjai, J.M. Melenk, R.H. Nochetto, E. Otárola, A.J. Salgado, and Ch. Schwab, Tensor FEM for spectral fractional diffusion. Found. Comput. Math. (2018), 1–62; 10.1007/s10208-018-9402-3.Search in Google Scholar

[6] M.Š. Birman and M.Z. Solomjak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve. Leningrad Univ., Leningrad (1980).Search in Google Scholar

[7] M. Bonforte, Y. Sire, and J.L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete Contin. Dyn. Syst., 35, No 12 (2015), 5725–5767.10.3934/dcds.2015.35.5725Search in Google Scholar

[8] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, No 5 (2010), 2052–2093.10.1016/j.aim.2010.01.025Search in Google Scholar

[9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations32, No 7-9 (2007), 1245–1260.10.1080/03605300600987306Search in Google Scholar

[10] A. Capella, J. Dávila, L. Dupaigne, and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Partial Differential Equations. 36, No 8 (2011), 1353–1384.10.1080/03605302.2011.562954Search in Google Scholar

[11] A. de Pablo, F. Quirós, A. Rodríguez, and J.L. Vázquez, A fractional porous medium equation. Adv. Math. 226, No 2 (2011), 1378–1409.10.1016/j.aim.2010.07.017Search in Google Scholar

[12] A. de Pablo, F. Quirós, A. Rodríguez, and J.L. Vázquez, A general fractional porous medium equation. Comm. Pur. Appl. Math. 65, No 9 (2012), 1242–1284.10.1002/cpa.21408Search in Google Scholar

[13] M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation. Numer. Algorithms73, No 2 (2016), 445–476.10.1007/s11075-016-0103-1Search in Google Scholar

[14] R. Du, W.R. Cao, and Z.Z. Sun, A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34, No 10 (2010), 2998–3007.10.1016/j.apm.2010.01.008Search in Google Scholar

[15] J. Duoandikoetxea, Fourier Analysis. Ser. Graduate Studies in Mathematics # 29, American Mathematical Society, Providence - RI (2001).Search in Google Scholar

[16] L.C. Evans, Partial Differential Equations. Ser. Graduate Studies in Mathematics # 19, American Mathematical Society, Providence - RI, 2nd Ed. (2010).Search in Google Scholar

[17] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad. 43 (1967), 82–86.10.3792/pja/1195521686Search in Google Scholar

[18] V. Gol’dshtein and A. Ukhlov, Weighted Sobolev spaces and embedding theorems. Trans. Amer. Math. Soc. 361, No 7 (2009), 3829–3850.10.1090/S0002-9947-09-04615-7Search in Google Scholar

[19] R. Gorenflo, A.A. Kilbas, F. Mainardi, and S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer Monographs in Mathematics, Springer, Heidelberg (2014).10.1007/978-3-662-43930-2Search in Google Scholar

[20] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Ser. Classics in Applied Mathematics, # 69, SIAM, Philadelphia, PA (2011).10.1137/1.9781611972030Search in Google Scholar

[21] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs. The Clarendon Press - Oxford Univ. Press, New York (1993).Search in Google Scholar

[22] V.R. Hosseini, E. Shivanian, and W. Chen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping. J. Comput. Phys. 312 (2016), 307–332.10.1016/ in Google Scholar

[23] X. Hu and L. Zhang, On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems. Appl. Math. Comput. 218, No 9 (2012), 5019–5034.10.1016/j.amc.2011.10.069Search in Google Scholar

[24] J.-P. Kahane, Teoría constructiva de funciones. Universidad de Buenos Aires, Buenos Aires (1961).Search in Google Scholar

[25] Y. Kian and M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations. Fract. Calc. Appl. Anal., 20, No 1 (2017), 117–138; 10.1515/fca-2017-0006; in Google Scholar

[26] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006).Search in Google Scholar

[27] A. Kufner. Weighted Sobolev Spaces. Ser. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1980).Search in Google Scholar

[28] A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25, No 3 (1984), 537–554.Search in Google Scholar

[29] N.S. Landkof, Foundations of Modern Potential Theory. Springer-Verlag, New York-Heidelberg (1972).10.1007/978-3-642-65183-0Search in Google Scholar

[30] L. Li, D. Xu, and M. Luo, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255 (2013), 471–485.10.1016/ in Google Scholar

[31] Z. Li, O.Y. Imanuvilov, and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations. Inverse Probl. 32, No 1 (2016).10.1088/0266-5611/32/1/015004Search in Google Scholar

[32] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I. Springer-Verlag, New York-Heidelberg (1972).10.1007/978-3-642-65161-8Search in Google Scholar

[33] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59, No 5 (2010), 1766–1772.10.1016/j.camwa.2009.08.015Search in Google Scholar

[34] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000).Search in Google Scholar

[35] W. McLean, Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, No 2 (2010), 123–138.10.1017/S1446181111000617Search in Google Scholar

[36] W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive-type memory term. J. Austral. Math. Soc. Ser. B35, No 1 (1993), 23–70.10.1017/S0334270000007268Search in Google Scholar

[37] D. Meidner, J. Pfefferer, K. Schürholz, and B. Vexler, hp-finite elements for fractional diffusion. SIA. J. Numer. Anal. 56, No 4 (2018), 2345–2374.10.1137/17M1135517Search in Google Scholar

[38] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–226.10.1090/S0002-9947-1972-0293384-6Search in Google Scholar

[39] R.H. Nochetto, E. Otárola, and A.J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15, No 3 (2015), 733–791.10.1007/s10208-014-9208-xSearch in Google Scholar

[40] F.W.J. Olver, Asymptotics and Special Functions. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1974).Search in Google Scholar

[41] E. Otárola, A PDE Approach to Numerical Fractional Diffusion. PhD thesis, University of Maryland, College Park (2014).Search in Google Scholar

[42] I. Podlubny, Fractional Differential Equations, Ser. Mathematics in Science and Engineering #198, Academic Press, Inc., San Diego, CA (1999).Search in Google Scholar

[43] T. Roubíček. Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics #153, Birkhäuser/Springer Basel AG, Basel, 2nd Ed. (2013).10.1007/978-3-0348-0513-1Search in Google Scholar

[44] 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, No 1 (2011), 426–447.10.1016/j.jmaa.2011.04.058Search in Google Scholar

[45] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science Publ., Yverdon (1993).Search in Google Scholar

[46] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60, No 1 (2007), 67–112.10.1002/cpa.20153Search in Google Scholar

[47] P.R. Stinga and J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Part. Diff. Eqs. 35, No 11 (2010), 2092–2122.10.1080/03605301003735680Search in Google Scholar

[48] Z.-Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, No 2 (2006), 193–209.10.1016/j.apnum.2005.03.003Search in Google Scholar

[49] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana # 3, Springer, Berlin (2007).Search in Google Scholar

[50] B.O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics # 1736, Springer-Verlag, Berlin (2000).10.1007/BFb0103908Search in Google Scholar

[51] J.L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type. J. Math. Pures Appl. (9), 101, No 5 (2014), 553–582.10.1016/j.matpur.2013.07.001Search in Google Scholar

[52] Y.-N. Zhang, Z.-Z. Sun, and X. Zhao. Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, No 3 (2012), 1535–1555.10.1137/110840959Search in Google Scholar

[53] X. Zhao and Z.-Z. Sun, Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, No 3 (2015), 747–771.10.1007/s10915-014-9874-5Search in Google Scholar

Received: 2017-10-16
Published Online: 2019-01-13
Published in Print: 2018-10-25

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