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The critical exponent for a time fractional diffusion equation with nonlinear memory
Author(s) -
Zhang Quanguo,
Li Yaning
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5169
Subject(s) - mathematics , exponent , nonlinear system , critical exponent , order (exchange) , diffusion , fractional calculus , mathematical analysis , mathematical physics , quantum mechanics , physics , geometry , philosophy , linguistics , finance , scaling , economics
In this paper, we determine the Fujita critical exponent of the following time fractional subdiffusion equation with nonlinear memory0 C D t α u − △ u =0 I t 1 − γ| u | p − 1 u , x ∈ R N , t > 0 ,u ( 0 , x ) = u 0 ( x ) , x ∈ R N ,where 0 < α < 1, 0 ≤ γ < 1, γ ≤ α , p > 1, u 0 ∈ C 0R N,0I t 1 − γdenotes left Riemann‐Liouville fractional integral of order 1 − γ . Let β = 1 − γ . We prove that, if 1 < p ≤ p ∗ = max 1 + 2 ( α + β )2 + α N − 2 ( α + β )+, 1 γ, any nontrivial positive solution blows up in a finite time. If p > p ∗ and ‖ u 0‖L q cR Nis sufficiently small, where q c = N α ( p − 1 ) 2 ( α + β ) , then u exists globally.