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Blow‐up phenomenon and the exact blow‐up time for a class of pseudo‐parabolic equations with nonlocal source
Author(s) -
Khelghati Ali,
Baghaei Khadijeh
Publication year - 2020
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.6841
Subject(s) - mathematics , bounded function , mathematical analysis , domain (mathematical analysis) , limiting , parabolic partial differential equation , energy (signal processing) , integrable system , function (biology) , class (philosophy) , boundary (topology) , phenomenon , partial differential equation , physics , mechanical engineering , statistics , evolutionary biology , engineering , biology , quantum mechanics , artificial intelligence , computer science
This paper deals with the blow‐up phenomenon to the following quasi‐linear pseudo‐parabolic equation with nonlocal source:u t − Δ u t − ∇ · ( | ∇ u | 2 q ∇ u ) = u p ( x , t ) ∫ Ω K ( x , y ) u p + 1 ( y , t ) d y , x , y ∈ Ω , t > 0 , where Ω ⊆ ℝ n , n ≥ 3 , is a bounded domain with smooth boundary. Here, 0 < q ≤ p and K ( x , y ) is an integrable real‐valued function. We show that for q < p , the blow‐up occurs in finite time with suitable initial data and arbitrary positive initial energy. We also state some key results based on the conception of limiting the energy function in the case of nonnegative initial energy. Besides, we obtain the exact blow‐up time under some certain conditions.