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 . As applications of the weak Harnack inequality we establish the strong maximum principle, the continuity of weak solutions at , and a uniqueness theorem for global bounded weak solutions.
@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/
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