Decay estimates for time-fractional and other non-local in time subdiffusion equations in \(\mathbb R^d\). (English) Zbl 1354.35178

The authors study the asymptotic behavior of the solution to the equation \[ \partial_t(k*[u-u_0]) - \Delta u=0, \quad t >0, \quad x\in \mathbb R^d, \] where \(u_{|t=0}= u_0\) and \(k\) is locally integrable, non-negative, and non-increasing with \(\lim_{t\downarrow 0} k(t)=+\infty\). The most important special case that is given most attention is the case of a fractional derivative of order \(\alpha\), i.e., \(k(t)=\frac 1{\Gamma(1-\alpha)}t^{-\alpha}\). In addition they study the case where the Laplacian \(\Delta\) is replaced by a more general elliptic operator in divergence form \(\div(A(t,x)\nabla u)\) with merely measurable, uniformly elliptic coefficient matrix \(A\).
For the case of the fractional derivative of order \(\alpha\) the authors obtain quite precise results, e.g. \(|u(t,\cdot)|_2 \lessapprox t^{-\min\{\frac {\alpha d}4,\alpha\}}\) when \(d\neq 4\) and \(|u(t,\cdot)|_{2,\infty} \lessapprox t^{-\alpha}\) when \(d= 4\) and thus they establish the existence of a critical dimension \(d=4\).
The techniques used involve the fundamental solution, Fourier multiplier methods and energy estimates.


35R11 Fractional partial differential equations
45K05 Integro-partial differential equations
47G20 Integro-differential operators
Full Text: DOI arXiv


[1] Bjorland, C; Schonbek, ME, Poincaré’s inequality and diffusive evolution equations, Adv. Differ. Equ., 14, 241-260, (2009) · Zbl 1169.35047
[2] Brändle, C., de Pablo, A.: Decay estimates for linear and nonlinear nonlocal heat equations. Available online at arXiv:1312.4661v1 · Zbl 1264.35271
[3] Bouchaud, J-P; Georges, A, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195, 127-293, (1990)
[4] Caffarelli, L; Vazquez, JL, Asymptotic behaviour of a porous medium equation with fractional diffusion, Discrete Contin. Dyn. Syst., 29, 1393-1404, (2011) · Zbl 1211.35043
[5] Caputo, M, Diffusion of fluids in porous media with memory, Geothermics, 28, 113-130, (1999)
[6] Chasseigne, E; Chaves, M; Rossi, JD, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86, 271-291, (2006) · Zbl 1126.35081
[7] Clément, P; Nohel, JA, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal., 10, 365-388, (1979) · Zbl 0411.45012
[8] Clément, P; Nohel, JA, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12, 514-534, (1981) · Zbl 0462.45025
[9] Dräger, J; Klafter, J, Strong anomaly in diffusion generated by iterated maps, Phys. Rev. Lett., 84, 5998-6001, (2000)
[10] Duoandikoetxea, J; Zuazua, E, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math., 315, 693-698, (1992) · Zbl 0755.45019
[11] Eidelman, SE; Kochubei, AN, Cauchy problem for fractional diffusion equations, J. Differ. Equ., 199, 211-255, (2004) · Zbl 1068.35037
[12] Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. John Wiley & Sons Inc, New York (1971) · Zbl 0219.60003
[13] Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, London (2004) · Zbl 1148.42001
[14] Gripenberg, G, Volterra integro-differential equations with accretive nonlinearity, J. Differ. Equ., 60, 57-79, (1985) · Zbl 0575.45013
[15] Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations. Encyclopedia of Mathematics and its Applications, vol. 34. Cambridge University Press, Cambridge (1990) · Zbl 0695.45002
[16] Ignat, LI; Rossi, JD, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl., 92, 163-187, (2009) · Zbl 1173.35363
[17] Jakubowski, V.G.: Nonlinear elliptic-parabolic integro-differential equations with \(L_1\)-data: existence, uniqueness, asymptotics. Dissertation, University of Essen (2001) · Zbl 1157.45309
[18] Kilbas, A.A., Saigo, M.: H-Transforms: Theory and Application. CRC Press, LLC, Boca Raton (2004) · Zbl 1056.44001
[19] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[20] Kochubei, AN, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340, 252-281, (2008) · Zbl 1149.26014
[21] Kochubei, AN, Fractional-order diffusion, Differ. Equ., 26, 485492, (1990) · Zbl 0729.35064
[22] Kochubei, AN, General fractional calculus, evolution equations, and renewal processes, Integr. Equ. Oper. Theory, 71, 583-600, (2011) · Zbl 1250.26006
[23] Ma, Y; Zhang, F; Changpin, L, The asymptotics of the solutions to the anomalous diffusion equations, Comput. Math. Appl., 66, 682-692, (2013) · Zbl 1345.35133
[24] Meerschaert, MM; Nane, E; Vellaisamy, P, Fractional Cauchy problems on bounded domains, Ann. Probab., 37, 979-1007, (2009) · Zbl 1247.60078
[25] Metzler, R; Klafter, J, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77, (2000) · Zbl 0984.82032
[26] Metzler, R; Klafter, J, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37, r161-r208, (2004) · Zbl 1075.82018
[27] Nakagawa, J., Sakamoto, K., Yamamoto, M.: Overview to mathematical analysis for fractional diffusion equations new mathematical aspects motivated by industrial collaboration. J. Math. Ind. 2(2010A-10), 99-108 (2010) · Zbl 1206.35247
[28] Nash, J, Continuity of solutions of parabolic and elliptic equations, Am. J. Math., 80, 931-954, (1958) · Zbl 0096.06902
[29] Prüss, J.: Evolutionary Integral Equations and Applications. Monographs in Mathematics, vol. 87. Birkhäuser, Basel (1993)
[30] Quittner, P., Souplet, P.: Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Birkhäuser Verlag, Basel (2007) · Zbl 1128.35003
[31] Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002) · Zbl 0991.35002
[32] Schiessel, H; Sokolov, IM; Blumen, A, Dynamics of a polyampholyte hooked around an obstacle, Phys. Rev. E, 56, r2390-r2393, (1997)
[33] Schilling, R., Song, R., Vondracek, Z.: Bernstein Functions. Theory and Applications. Studies in Mathematics, vol. 37. De Gruyter, Berlin (2010) · Zbl 1197.33002
[34] Schneider, WR; Wyss, W, Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144, (1989) · Zbl 0692.45004
[35] Sinai, YG, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl., 27, 256-268, (1982) · Zbl 0505.60086
[36] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd rev. and enl. edn. Johann Ambrosius Barth Verlag, Heidelberg (1995) · Zbl 0755.45019
[37] Uchaikin, V.V.: Fractional Derivatives for Physicists and Engineers. Volume 1 Background and Theory. Nonlinear Physical Science, Springer, Heidelberg (2013) · Zbl 1312.26002
[38] Vazquez, JL, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc., 16, 769-803, (2014) · Zbl 1297.35279
[39] Vergara, V; Zacher, R, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z., 259, 287-309, (2008) · Zbl 1144.45003
[40] Vergara, V; Zacher, R, Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods, SIAM J. Math. Anal., 47, 210-239, (2015) · Zbl 1317.45006
[41] Zacher, R, A De Giorgi-Nash type theorem for time fractional diffusion equations, Math. Ann., 356, 99-146, (2013) · Zbl 1264.35271
[42] Zacher, R, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348, 137-149, (2008) · Zbl 1154.45008
[43] Zacher, R, Maximal regularity of type \(L_p\) for abstract parabolic Volterra equations, J. Evol. Equ., 5, 79-103, (2005) · Zbl 1104.45008
[44] Zuazua, E.: Large time asymptotics for heat and dissipative wave equations. Manuscript available at http://www.uam.es/enrique.zuazua (2003)
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