六维能量临界热方程的一类 II 型爆破
偏微分方程分析
2020-02-04 v1
摘要
我们研究 R n × ( 0 , T ) \R^n\times(0,T) R n × ( 0 , T ) u t = Δ u + ∣ u ∣ p − 1 u u_t=\Delta u+|u|^{p-1}u u t = Δ u + ∣ u ∣ p − 1 u u ( x , t ) ≈ { λ ( t ) − 2 Q ( λ ( t ) − 1 x ) in the inner region: ∣ x ∣ ∼ λ ( t ) , − ( p − 1 ) 1 p − 1 ( T − t ) − 1 p − 1 in the selfsimilar region: ∣ x ∣ ∼ T − t u(x,t)\approx \begin{cases} \lambda(t)^{-2}{\sf Q}(\lambda(t)^{-1}x) & \text{in the inner region: } |x|\sim\lambda(t), -(p-1)^\frac{1}{p-1}(T-t)^{-\frac{1}{p-1}} & \text{in the selfsimilar region: } |x|\sim\sqrt{T-t} \end{cases} u ( x , t ) ≈ { λ ( t ) − 2 Q ( λ ( t ) − 1 x ) in the inner region: ∣ x ∣ ∼ λ ( t ) , − ( p − 1 ) p − 1 1 ( T − t ) − p − 1 1 in the selfsimilar region: ∣ x ∣ ∼ T − t λ ( t ) = ( 1 + o ( 1 ) ) ( T − t ) 5 4 ∣ log ( T − t ) ∣ − 15 8 \lambda(t)=(1+o(1))(T-t)^\frac{5}{4}|\log(T-t)|^{-\frac{15}{8}} λ ( t ) = ( 1 + o ( 1 )) ( T − t ) 4 5 ∣ log ( T − t ) ∣ − 8 15 E loc ( u ) = 1 2 ∥ ∇ u ∥ L 2 ( ∣ x ∣ < 1 ) 2 − 1 3 ∥ u ∥ L 3 ( ∣ x ∣ < 1 ) 3 E_\text{loc}(u) =\frac{1}{2}\|\nabla u\|_{L^2(|x|<1)}^2-\frac{1}{3}\|u\|_{L^3(|x|<1)}^3 E loc ( u ) = 2 1 ∥∇ u ∥ L 2 ( ∣ x ∣ < 1 ) 2 − 3 1 ∥ u ∥ L 3 ( ∣ x ∣ < 1 ) 3 − ∞ -\infty − ∞
引用
@article{arxiv.2002.00528,
title = {A type II blowup for the six dimensional energy critical heat equation},
author = {Junichi Harada},
journal= {arXiv preprint arXiv:2002.00528},
year = {2020}
}