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Lifespan for a semilinear pseudo‐parabolic equation
Author(s) -
Xu Guangyu,
Zhou Jun
Publication year - 2017
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.4639
Subject(s) - mathematics , bounded function , domain (mathematical analysis) , energy (signal processing) , pure mathematics , mathematical analysis , statistics
This paper deals with the blow‐up solution to the following semilinear pseudo‐parabolic equationu t = Δ u t + Δ u + | u | p − 1 u in a bounded domain Ω ⊂ R n , which was studied by Luo (Math Method Appl Sci 38(12):2636‐2641, 2015) with the following assumptions on p :1 < p < + ∞ ,if n = 1 , 2 ;1 < p ≤ n + 2 n − 2 ,if n ≥ 3 ,and the lifespan for the initial energy J ( u 0 )<0 is considered. This paper generalizes the above results on the following two aspects: a new blow‐up condition is given, which holds for all p> 1; a new lifespan is given, which holds for all p> 1 and possible J ( u 0 )≥0. Moreover, as a byproduct, we refine the lifespan when J ( u 0 )<0.