TY - JOUR
T1 - Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation
AU - Ghoul, Tej Eddine
AU - Nguyen, Van Tien
AU - Zaag, Hatem
N1 - Publisher Copyright:
© 2017 by De Gruyter.
PY - 2017/1/2
Y1 - 2017/1/2
N2 - We consider u (x,t) a solution of ∂tu = Δu + |u|p-1u which blows up at some time T > 0, where u : ℝN × [0,T) → ℝ, p > 1 and (N-2)p < N + 2. Define S ⊂ ℝN to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an (N-ℓ-dimensional continuum for some ℓϵ{1,...,N-1}, then S is in fact a 2 manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable (T-t) and reach significant small terms in the polynomial order (T-t)μ for some μ > 0. Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.
AB - We consider u (x,t) a solution of ∂tu = Δu + |u|p-1u which blows up at some time T > 0, where u : ℝN × [0,T) → ℝ, p > 1 and (N-2)p < N + 2. Define S ⊂ ℝN to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an (N-ℓ-dimensional continuum for some ℓϵ{1,...,N-1}, then S is in fact a 2 manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable (T-t) and reach significant small terms in the polynomial order (T-t)μ for some μ > 0. Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.
KW - Blow-Up Profile
KW - Blow-Up Set
KW - Blow-Up Solution
KW - Regularity
KW - Semilinear Heat Equation
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U2 - 10.1515/ans-2016-6005
DO - 10.1515/ans-2016-6005
M3 - Article
AN - SCOPUS:85011688987
SN - 1536-1365
VL - 17
SP - 31
EP - 54
JO - Advanced Nonlinear Studies
JF - Advanced Nonlinear Studies
IS - 1
ER -