TY - JOUR
T1 - Upper estimates for blow-up solutions of a quasi-linear parabolic equation
AU - Anada, Koichi
AU - Ishiwata, Tetsuya
AU - Ushijima, Takeo
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2024/1
Y1 - 2024/1
N2 - In this paper, we consider a quasi-linear parabolic equation ut= up(xxx+ u) . It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p> 2 . We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2 < p< 3 and in the case of p= 3 , separately. In addition, we assert that if 2 ≤ p≤ 3 then limt↗T(T-t)1p+εmaxu(x,t)=0 z for any ε> 0 under some assumptions.
AB - In this paper, we consider a quasi-linear parabolic equation ut= up(xxx+ u) . It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p> 2 . We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2 < p< 3 and in the case of p= 3 , separately. In addition, we assert that if 2 ≤ p≤ 3 then limt↗T(T-t)1p+εmaxu(x,t)=0 z for any ε> 0 under some assumptions.
KW - Asymptotic behavior
KW - Blow-up phenomena
KW - Quasi-linear parabolic equation
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U2 - 10.1007/s13160-023-00606-6
DO - 10.1007/s13160-023-00606-6
M3 - Article
AN - SCOPUS:85167342307
SN - 0916-7005
VL - 41
SP - 381
EP - 405
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
IS - 1
ER -