On a connection between powers of operators and fractional Cauchy problems. (English) Zbl 1336.47046
The paper is devoted to connections between ordinary non-homogeneous equations and fractional abstract Cauchy problems (FACP) on Banach spaces. In particular, by using the subordination principle, the authors provide an explicit form of the common solution involving special functions whenever the domination operator of FACP is the generator of a \(C_0\)-semigroup. Some illustrations of the main result are also given at the end of the paper.
Reviewer: Chuang Chen (Chengdu)
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 35C05 | Solutions to PDEs in closed form |
| 60J35 | Transition functions, generators and resolvents |
| 26A33 | Fractional derivatives and integrals |
| 35K90 | Abstract parabolic equations |
| 34G10 | Linear differential equations in abstract spaces |
Keywords:
perturbed fourth order PDEs; Cauchy problem; iterated Brownian motion; Caputo derivative; \(C_0\)-semigroups; PDE connection; \(\alpha\)-resolvent families; Wright functionReferences:
| [1] | Allouba H., Zheng W.: Brownian-time processes: The PDE connection and the halfderivative generator. Ann. Prob. 29, 1780–1795 (2001) · Zbl 1018.60066 · doi:10.1214/aop/1015345772 |
| [2] | Allouba H.: Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula Trans. Amer. Math. Soc. 354(11), 4627–4637 (2002) · Zbl 1006.60063 · doi:10.1090/S0002-9947-02-03074-X |
| [3] | Arendt W., Batty C., Hieber M., Neubrander F.: Vector-valued Laplace Transforms and Cauchy Problems Monographs in Mathematics, 96. Birkhäuser, Basel (2001) · Zbl 0978.34001 |
| [4] | Baeumer B., Meerschaert M.M., Nane E.: Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc., 361(7), 3915–3930 (2009) · Zbl 1186.60079 · doi:10.1090/S0002-9947-09-04678-9 |
| [5] | E. Bazhlekova. Fractional Evolution Equations in Banach Spaces Ph.D. Thesis, Eindhoven University of Technology, 2001. |
| [6] | Carasso A. S., Kato T.: On subordinated holomorphic semigroups. Trans. Amer. Math. Soc. 327(2), 867–878 (1991) · Zbl 0743.47017 · doi:10.1090/S0002-9947-1991-1018572-4 |
| [7] | Ph. Clément, G. Gripenberg, S.-O. Londen. Schauder estimates for equations with fractional derivatives. Trans. Amer. Math. Soc. 352 (2000), 2239–2260. · Zbl 0947.35023 |
| [8] | DeBlassie D.: Iterated Brownian motion in an open set. Ann. Prob. 14(3), 1529–1558 (2004) · Zbl 1051.60082 · doi:10.1214/105051604000000404 |
| [9] | K.J.-Engel, R. Nagel. One-parameter Semigroups for Linear Evolution Equations Graduate Texts in Math.,194, Springer, New York, 2000. · Zbl 0952.47036 |
| [10] | K. Fujii. A higher order non-linear differential equation and a generalization of the Airy function Preprint. Available at arXiv:0712.2481v1 |
| [11] | Gorenflo R., Luchko Y., Mainardi F.: Analytical properties and applications of the Wright function. Fractional Calculus and Applied Analysis, 2(4), 383–414 (1999) · Zbl 1027.33006 |
| [12] | Kulish V.V., Lage J.L.: Application of fractional calculus to fluid mechanics. Journal of Fluids Engineering, 124, 803–806 (2002) · doi:10.1115/1.1478062 |
| [13] | Lizama C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000) · Zbl 0952.45005 · doi:10.1006/jmaa.1999.6668 |
| [14] | Nane E.: Higher order PDE’s and iterated processes. Trans. Amer. Math. Soc. 360(5), 2681–2692 (2008) · Zbl 1157.60071 · doi:10.1090/S0002-9947-07-04437-6 |
| [15] | P. Miller. Applied Asymptotic Analysis Graduate Studies in Mathematics, 75, 2006. |
| [16] | J. Prüss. Evolutionary Integral Equations and Applications, Monographs Math., 87, Birkhäuser, Verlag, 1993. |
| [17] | Wright E. M.: The generalized Bessel function of order greater than one. Quarterly Journal of Mathematics (Oxford ser.) 11, 36–48 (1940) · Zbl 0023.14101 · doi:10.1093/qmath/os-11.1.36 |
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