Singular solutions for space-time fractional equations in a bounded domain
| dc.contributor.author | Chan, Hardy | |
| dc.contributor.author | Gómez Castro, David | |
| dc.contributor.author | Vázquez, Juan Luis | |
| dc.date.accessioned | 2023-06-22T12:55:03Z | |
| dc.date.available | 2023-06-22T12:55:03Z | |
| dc.date.issued | 2023-04-11 | |
| dc.description.abstract | This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann-Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense. | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.faculty | Instituto de Matemática Interdisciplinar (IMI) | |
| dc.description.refereed | FALSE | |
| dc.description.sponsorship | Unión Europea | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovación | |
| dc.description.sponsorship | Swiss National Science Foundation | |
| dc.description.status | unpub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/77760 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/73324 | |
| dc.language.iso | eng | |
| dc.relation.projectID | No. 883363 | |
| dc.relation.projectID | PID2021-127105NB-I00; PGC2018-098440-B-I00 | |
| dc.relation.projectID | PZ00P2 - 202012/1 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 517.95 | |
| dc.subject.keyword | Analysis of PDEs | |
| dc.subject.ucm | Análisis matemático | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202 Análisis y Análisis Funcional | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | Singular solutions for space-time fractional equations in a bounded domain | |
| dc.type | journal article | |
| dcterms.references | [1] N. Abatangelo. Large s-Harmonic Functions and Boundary Blow-up Solutions for the Fractional Laplacian. Discrete and Continuous Dynamical Systems 35.12 (2015), pp. 5555–5607. [2] N. Abatangelo and L. Dupaigne. Nonhomogeneous Boundary Conditions for the Spectral Fractional Laplacian. Ann. l’Institut Henri Poincare Anal. Non Lineaire 34.2 (2017), pp. 439–467. arXiv: 1509.06275. [3] N. Abatangelo, D. Gómez-Castro, and J. L. Vázquez. Singular boundary behaviour and large solutions for fractional elliptic equations (2019), pp. 1–42. arXiv: 1910.00366. [4] K. Bogdan, T. Grzywny, and M. Ryznar. Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Annals of Probability 38.5 (2010), pp. 1901–1923. [5] M. Bonforte, A. Figalli, and J. L. Vázquez. Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. PDE 11.4 (2018), pp. 945–982. [6] H. Chan, D. Gómez-Castro, and J. L. Vázquez. Singular solutions for fractional parabolic boundary value problems (2020). arXiv: 2007.13391. [7] Z. Q. Chen, P. Kim, and R. Song. Heat kernel estimates for the Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12.5 (2010), pp. 1307–1327. [8] Z. Q. Chen, P. Kim, and R. Song. Two-sided heat kernel estimates for censored stable-like processes. Probability Theory and Related Fields 146.3 (2009), pp. 361–399. [9] C. Cortázar, F. Quirós, and N. Wolanski. A heat equation with memory: large-time behavior. J. Funct. Anal. 281.9 (2021), Paper No. 109174, 40. [10] C. Cortázar, F. Quirós, and N. Wolanski. Asymptotic profiles for inhomogeneous heat equations with memory. 2023. [11] C. G. Gal and M. Warma. Fractional-in-Time Semilinear Parabolic Equations and Applications. Vol. 84. Mathématiques et Applications August. Cham: Springer International Publishing, 2020. [12] A. Kilbas, M. Rivero, L. Rodríguez-Germá, and J. Trujillo. Caputo linear fractional differential equations. IFAC Proceedings Volumes 39.11 (2006), pp. 52–57. [13] F. Mainardi. On some properties of the Mittag-Leffler function Eα(−t α) completely monotone for t > 0 and 0 < α < 1. Discrete & Continuous Dynamical Systems - B 19.7 (2014), pp. 2267–2278. [14] F. Mainardi, A. Mura, and G. Pagnini. The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. International Journal of Differential Equations 2010 (Feb. 2010), e104505. [15] R. Song. Sharp bounds on the density, Green function and jumping function of subordinate killed BM. Probab. Theory Relat. Fields 128.4 (2004), pp. 606–628. [16] R. Song, L. Xie, and Y. Xie. Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients. Science China Mathematics 63.11 (2020), pp. 2343–2362. arXiv: 1712.07565. | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | fe4c14a6-9c7a-4ce9-a868-b42d84aada05 | |
| relation.isAuthorOfPublication.latestForDiscovery | fe4c14a6-9c7a-4ce9-a868-b42d84aada05 |
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