A weak Harnack inequality for fractional evolution equations with discontinuous coefficients
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 903-940.

We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As a main result we establish for nonnegative weak supersolutions of such problems a weak Harnack inequality with optimal critical exponent. The proof relies on new a priori estimates for time fractional problems and uses Moser’s iteration technique and an abstract lemma of Bombieri and Giusti, the latter allowing to avoid the rather technically involved approach via BMO. As applications of the weak Harnack inequality we establish the strong maximum principle, the continuity of weak solutions at t=0, and a uniqueness theorem for global bounded weak solutions.

Publié le :
Classification : 35R09, 45K05
@article{ASNSP_2013_5_12_4_903_0,
     author = {Zacher, Rico},
     title = {A weak {Harnack} inequality for fractional evolution equations with discontinuous coefficients},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {903--940},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
     mrnumber = {3184573},
     zbl = {1285.35124},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_903_0/}
}
TY  - JOUR
AU  - Zacher, Rico
TI  - A weak Harnack inequality for fractional evolution equations with discontinuous coefficients
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2013
SP  - 903
EP  - 940
VL  - 12
IS  - 4
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2013_5_12_4_903_0/
LA  - en
ID  - ASNSP_2013_5_12_4_903_0
ER  - 
%0 Journal Article
%A Zacher, Rico
%T A weak Harnack inequality for fractional evolution equations with discontinuous coefficients
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2013
%P 903-940
%V 12
%N 4
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2013_5_12_4_903_0/
%G en
%F ASNSP_2013_5_12_4_903_0
Zacher, Rico. A weak Harnack inequality for fractional evolution equations with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 903-940. http://www.numdam.org/item/ASNSP_2013_5_12_4_903_0/

[1] E. Bazhlekova, “Fractional Evolution Equations in Banach Spaces”, Dissertation, Technische Universiteit Eindhoven, 2001. | MR | Zbl

[2] E. Bombieri and E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Invent. Math. 15 (1972), 24–46. | EuDML | MR | Zbl

[3] Ph. Clément, On abstract Volterra equations in Banach spaces with completely positive kernels, In: “Infinite-dimensional systems” (Retzhof, 1983), Lecture Notes in Math., Vol. 1076, Springer, Berlin, 1984, 32–40. | MR | Zbl

[4] Ph. Clément, S.-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Equations 196 (2004), 418–447. | MR | Zbl

[5] Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal. 12 (1981), 514–534. | MR | Zbl

[6] Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73–105. | EuDML | MR | Zbl

[7] Ph. Clément and J. Prüss, Global existence for a semilinear parabolic Volterra equation, Math. Z. 209 (1992), 17–26. | EuDML | MR | Zbl

[8] Ph. Clément and R. Zacher, A priori estimates for weak solutions of elliptic equations, Technical Report, Martin-Luther University Halle-Wittenberg, Germany, 2004.

[9] E. DiBenedetto, “Degenerate Parabolic Equations”, Springer, New York, 1993. | MR | Zbl

[10] S. E. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), 211–255. | MR | Zbl

[11] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer, 1977. | MR | Zbl

[12] G. Gripenberg, Volterra integro-differential equations with accretive nonlinearity, J. Differential Equations 60 (1985), 57–79. | MR | Zbl

[13] G. Gripenberg, S.-O. Londen and O. Staffans, “Volterra Integral and Functional Equations”, In: Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. | MR | Zbl

[14] R. Hilfer, Fractional time evolution, In: “Applications of Fractional Calculus in Physics”, R. Hilfer (ed.), World Sci. Publ., River Edge, NJ, 2000, 87–130. | MR | Zbl

[15] R. Hilfer, On fractional diffusion and continuous time random walks, Phys. A 329 (2003), 35–40. | MR | Zbl

[16] M. Kassmann, The classical Harnack inequality fails for non-local operators, SFB 611-preprint no. 360, University of Bonn, Germany, 2007.

[17] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations”, Elsevier, 2006. | MR | Zbl

[18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, “Linear and Quasilinear Equations of Parabolic Type”, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. | MR | Zbl

[19] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific, London, 1996. | MR | Zbl

[20] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77. | MR | Zbl

[21] J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. | MR | Zbl

[22] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134; correction in Comm. Pure Appl. Math. 20 (1967), 231–236. | MR | Zbl

[23] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727–740. | MR | Zbl

[24] J. Prüss, “Evolutionary Integral Equations and Applications”, Monographs in Mathematics, Vol.  87, Birkhäuser, Basel, 1993. | MR | Zbl

[25] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A 27 (1994), 3407–3410. | MR | Zbl

[26] L. Saloff-Coste, “Aspects of Sobolev-Type Inequalities”, London Mathematical Society Lecture Note Series, Vol. 289, University Press, Cambridge, 2002. | MR | Zbl

[27] E. Scalas, R. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A 284 (2000), 376–384. | MR | Zbl

[28] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), 1155–1174. | MR | Zbl

[29] N. S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. | MR | Zbl

[30] V. Vergara and R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008), 287–309. | MR | Zbl

[31] R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations, Math. Ann. 356 (2013), 99–146. | MR | Zbl

[32] R. Zacher, A weak Harnack inequality for fractional differential equations, J. Integral Equations Appl. 19 (2007), 209–232. | MR | Zbl

[33] R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl. 348 (2008), 137–149. | MR | Zbl

[34] R. Zacher, Maximal regularity of type L p for abstract parabolic Volterra equations, J. Evol. Equ. 5 (2005), 79–103. | MR | Zbl

[35] R. Zacher, Quasilinear parabolic integro-differential equations with nonlinear boundary conditions, Differential Integral Equations 19 (2006), 1129–1156. | MR | Zbl

[36] R. Zacher, The Harnack inequality for the Riemann-Liouville fractional derivation operator, Math. Inequal. Appl. 14 (2011), 35–43. | MR | Zbl

[37] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac. 52 (2009), 1–18. | MR | Zbl