Decay estimates for evolutionary equations with fractional time-diffusion. (English) Zbl 1461.35049

Summary: We consider an evolution equation whose time-diffusion is of fractional type, and we provide decay estimates in time for the \(L^s\)-norm of the solutions in a bounded domain. The spatial operator that we take into account is very general and comprises classical local and nonlocal diffusion equations.


35B40 Asymptotic behavior of solutions to PDEs
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35K90 Abstract parabolic equations
47J35 Nonlinear evolution equations
58D25 Equations in function spaces; evolution equations
Full Text: DOI arXiv


[1] Abatangelo, N.; Valdinoci, E., A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35, 793-815, (2014) · Zbl 1296.53021
[2] N. Abatangelo, E. Valdinoci, Getting acquainted with the fractional Laplacian. Springer INdAM Ser., Springer, Cham, 2019.
[3] E. Affili, E. Valdinoci, Decay estimates for evolution equations with classical and fractional time-derivatives. J. Differential Equations, in press. https://doi.org/10.1016/j.jde.2018.09.031.
[4] M. Allen, Uniqueness for weak solutions of parabolic equations with a fractional time derivative. Contemporary Mathematics, in press.
[5] Allen, M.; Caffarelli, L.; Vasseur, A., A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221, 603-630, (2016) · Zbl 1338.35428
[6] Arendt, W.; Prüss, J., Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. Math. Anal., 23, 412-448, (1992) · Zbl 0765.45009
[7] A. Atangana, A. Kilicman, On the generalized mass transport equation to the concept of variable fractional derivative. Math. Probl. Eng. 2014, Art. ID 542809, 9 pp.
[8] Barrios, B.; Figalli, A.; Valdinoci, E., Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13, 609-639, (2014) · Zbl 1316.35061
[9] Bhrawy, AH; Zaky, MA, An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations, Appl. Numer. Math., 111, 197-218, (2017) · Zbl 1353.65106
[10] C. Bucur, E. Valdinoci, Nonlocal diffusion and applications. Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. xii+155 pp.
[11] X. Cabré, M. M. Fall, J. Solà-Morales, T. Weth, Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay. J. Reine Angew. Math., in press. https://doi.org/10.1515/crelle-2015-0117.
[12] Cabré, X.; Serra, J., An extension problem for sums of fractional Laplacians and \(1\)-D symmetry of phase transitions, Nonlinear Anal., 137, 246-265, (2016) · Zbl 1386.35430
[13] Caffarelli, L.; Roquejoffre, J-M; Savin, O., Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63, 1111-1144, (2010) · Zbl 1248.53009
[14] Caputo, M., Linear Models of Dissipation whose \( Q\) is almost Frequency Independent-II, Geoph. J. Intern., 13, 529-539, (1967)
[15] Chambolle, A.; Morini, M.; Ponsiglione, M., A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal., 44, 4048-4077, (2012) · Zbl 1270.35019
[16] Chambolle, A.; Novaga, M.; Ruffini, B., Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound., 19, 393-415, (2017) · Zbl 1380.53070
[17] Chen, M.; Deng, W., A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation, Appl. Math. Lett., 68, 87-93, (2017) · Zbl 1361.65100
[18] Chen, T.; Liu, W., An anti-periodic boundary value problem for the fractional differential equation with a \(p\)-Laplacian operator, Appl. Math. Lett., 25, 1671-1675, (2012) · Zbl 1248.35219
[19] Cinti, E.; Sinestrari, C.; Valdinoci, E., Neckpinch singularities in fractional mean curvature flows, Proc. Amer. Math. Soc., 146, 2637-2646, (2018) · Zbl 1390.53068
[20] Ciraolo, G.; Figalli, A.; Maggi, F.; Novaga, M., Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, J. Reine Angew. Math., 741, 275-294, (2018) · Zbl 1411.53030
[21] Clément, Ph; Nohel, JA, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal., 10, 365-388, (1979) · Zbl 0411.45012
[22] Clément, Ph; Nohel, JA, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12, 514-535, (1981) · Zbl 0462.45025
[23] Cozzi, M.; Passalacqua, T., One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels, J. Differential Equations, 260, 6638-6696, (2016) · Zbl 1344.49001
[24] Dávila, J.; Pino, M.; Dipierro, S.; Valdinoci, E., Nonlocal Delaunay surfaces, Nonlinear Anal., 137, 357-380, (2016) · Zbl 1341.53006
[25] Pablo, A.; Quirós, F.; Rodríguez, A.; Vázquez, JL, A fractional porous medium equation, Adv. Math., 226, 1378-1409, (2011) · Zbl 1208.26016
[26] Pablo, A.; Quirós, F.; Rodríguez, A.; Vázquez, JL, A general fractional porous medium equation, Comm. Pure Appl. Math., 65, 1242-1284, (2012) · Zbl 1248.35220
[27] E. DiBenedetto, J. M. Urbano, V. Vespri, Current issues on singular and degenerate evolution equations. Evolutionary equations. Vol. I, 169-286, Handb. Differ. Equ., North-Holland, Amsterdam, 2004.
[28] Castro, A.; Kuusi, T.; Palatucci, G., Nonlocal Harnack inequalities, J. Funct. Anal., 267, 1807-1836, (2014) · Zbl 1302.35082
[29] S. Dipierro, E. Valdinoci, Nonlocal minimal surfaces: interior regularity, quantitative estimates and boundary stickiness. Recent developments in nonlocal theory, 165-209, De Gruyter, Berlin, 2018. · Zbl 1406.49048
[30] Ezzat, MA; Karamany, AS, Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures, Z. Angew. Math. Phys., 62, 937-952, (2011) · Zbl 1264.74049
[31] Farina, A.; Valdinoci, E., Regularity and rigidity theorems for a class of anisotropic nonlocal operators, Manuscripta Math., 153, 53-70, (2017) · Zbl 06714544
[32] A. Farina, E. Valdinoci, Flatness results for nonlocal minimal cones and subgraphs. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), in press.
[33] W. Feller, An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
[34] Figalli, A.; Valdinoci, E., Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., 729, 263-273, (2017) · Zbl 1380.49060
[35] E. Giusti, Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. xii+240 pp. · Zbl 0545.49018
[36] Gripenberg, G., Two Tauberian theorems for nonconvolution Volterra integral operators, Proc. Amer. Math. Soc., 89, 219-225, (1983) · Zbl 0533.45007
[37] Gripenberg, G., Volterra integro-differential equations with accretive nonlinearity, J. Differential Equations, 60, 57-79, (1985) · Zbl 0575.45013
[38] Iannizzotto, A.; Mosconi, S.; Squassina, M., Global Hölder regularity for the fractional \(p\)-Laplacian, Rev. Mat. Iberoam., 32, 1353-1392, (2016) · Zbl 1433.35447
[39] Kemppainen, J.; Siljander, J.; Vergara, V.; Zacher, R., Decay estimates for time-fractional and other non-local in time subdiffusion equations in \(\mathbb{R}^d\), Math. Ann., 366, 941-979, (2016) · Zbl 1354.35178
[40] Kemppainen, J.; Siljander, J.; Zacher, R., Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations, 263, 149-201, (2017) · Zbl 1366.35218
[41] J. Kemppainen, R. Zacher, Long-time behaviour of non-local in time Fokker-Planck equations via the entropy method. Preprint, arXiv:1708.04572 (2017).
[42] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. xvi+523.
[43] Kuusi, T.; Mingione, G.; Sire, Y., Nonlocal self-improving properties, Anal. PDE, 8, 57-114, (2015) · Zbl 1317.35284
[44] N. S Landkof, Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
[45] Li, Y.; Qi, A., Positive solutions for multi-point boundary value problems of fractional differential equations with \(p\)-Laplacian, Math. Methods Appl. Sci., 39, 1425-1434, (2016) · Zbl 1338.34018
[46] Liu, X.; Jia, M.; Ge, W., The method of lower and upper solutions for mixed fractional four-point boundary value problem with \(p\)-Laplacian operator, Appl. Math. Lett., 65, 56-62, (2017) · Zbl 1357.34018
[47] Luchko, Y.; Yamamoto, M., General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems, Fract. Calc. Appl. Anal., 19, 676-695, (2016) · Zbl 06604016
[48] Mainardi, F., On some properties of the Mittag-Leffler function \(E_\alpha (-t^\alpha )\), completely monotone for \(t >0\) with \(0<\alpha <1\), Discrete Contin. Dyn. Syst. Ser. B, 19, 2267-2278, (2014) · Zbl 1303.26007
[49] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4, 153-192, (2001) · Zbl 1054.35156
[50] Meerschaert, MM; Nane, E.; Vellaisamy, P., Fractional Cauchy problems on bounded domains, Ann. Probab., 37, 979-1007, (2009) · Zbl 1247.60078
[51] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 77, (2000) · Zbl 0984.82032
[52] Nakagawa, J.; Sakamoto, K.; Yamamoto, M., Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated by industrial collaboration, J. Math-for-Ind., 2A, 99-108, (2010) · Zbl 1206.35247
[53] Paris, R. B., Exponential asymptotics of the Mittag-Leffler function, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458, 3041-3052, (2002) · Zbl 1049.33018
[54] Pincherle, S., Sull’inversione degli integrali definiti, Memorie di Matem. e Fis. della Società italiana delle Scienze, Serie, 3, 3-43, (1907)
[55] Podio-Guidugli, P., A notion of nonlocal Gaussian curvature, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27, 181-193, (2016) · Zbl 1348.49044
[56] J. Prüss, Evolutionary integral equations and applications. Monographs in Mathematics, 87. Birkhäuser Verlag, Basel, 1993. xxvi+366 pp.
[57] P. Pucci, M. Xiang, B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional\(p\)-Laplacian in\({\mathbb{R}}^{N}\). Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785-2806. · Zbl 1329.35338
[58] Raviart, PA, Sur la résolution de certaines équations paraboliques non linéaires, J. Functional Analysis, 5, 299-328, (1970) · Zbl 0199.42401
[59] Ros-Oton, X.; Serra, J., Regularity theory for general stable operators, J. Differential Equations, 260, 8675-8715, (2016) · Zbl 1346.35220
[60] Ros-Oton, X.; Serra, J.; Valdinoci, E., Pohozaev identities for anisotropic integro-differential operators, Comm. Partial Differential Equations, 42, 1290-1321, (2017) · Zbl 1381.35048
[61] Ros-Oton, X.; Valdinoci, E., The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math., 288, 732-790, (2016) · Zbl 1334.35397
[62] M. Sáez and E. Valdinoci, On the evolution by fractional mean curvature. Comm. Anal. Geom. 27 (2019), no. 1.
[63] Sánchez, J.; Vergara, V., Long-time behavior of nonlinear integro-differential evolution equations, Nonlinear Anal., 91, 20-31, (2013) · Zbl 1297.45012
[64] L. E. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Ph.D. Thesis, The University of Texas at Austin, 2005.
[65] Topp, E.; Yangari, M., Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262, 6018-6046, (2017) · Zbl 1386.35192
[66] V. V. Uchaikin, Fractional derivatives for physicists and engineers. Volume II. Applications. Nonlinear Physical Science. Higher Education Press, Beijing; Springer, Heidelberg, 2013. xii+446 pp.
[67] J. L. Vázquez, The porous medium equation. Mathematical theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. xxii+624 pp.
[68] Vergara, V.; Zacher, R., A priori bounds for degenerate and singular evolutionary partial integro-differential equations, Nonlinear Anal., 73, 3572-3585, (2010) · Zbl 1205.45011
[69] Vergara, V.; Zacher, R., Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47, 210-239, (2015) · Zbl 1317.45006
[70] Volterra, V., Sopra alcune questioni di inversione di integrali definiti, Ann. Mat. Pura Appl., 25, 139-178, (1897) · JFM 28.0366.02
[71] Zacher, R., Maximal regularity of type \( L_p\) for abstract parabolic Volterra equations, J. Evol. Equ., 5, 79-103, (2005) · Zbl 1104.45008
[72] Zacher, R., Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac., 52, 1-18, (2009) · Zbl 1171.45003
[73] X. Zhang, L. Liu, B. Wiwatanapataphee, Y. Wu, The eigenvalue for a class of singular\(p\)-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Appl. Math. Comput. 235 (2014), 412-422.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.