论文标题
六维能量临界热方程的II型爆炸
A type II blowup for the six dimensional energy critical heat equation
论文作者
论文摘要
我们研究6D能量关键热方程的爆炸解决方案$ u_t =ΔU+| u |^{p-1} u $ in $ \ r^n \ times（0，t）$。本文的目的是显示Filippas，Herrero和Velázquez\ Cite {Filippashv}预测的II型爆炸解决方案的存在。尺寸六是边界案例，无论是否可能发生II型爆炸。因此，解决方案的行为与其他情况大不相同。实际上，我们的解决方案的行为就像 \ [ u（x，t）\大约 \ begin {case} λ（t）^{ - 2} {\ sf q}（λ（t）^{ - 1} x）＆\ text {在内部区域：} | x | \simλ（t）， - （p-1）^\ frac {1} {p-1}（t-t）^{ - \ frac {1} {p-1}}}＆\ text {in selfsimilar region：} | x | \ sim \ sqrt {t-t} \ end {cases} \]带$λ（t）=（1+o（1））（t-t）^\ frac {5} {4} {4} | \ log（t-t）|^{ - \ frac {15} {8}}} $。本地能量$ e_ \ text {loc}（u） = \ frac {1} {2} \ | \ nabla u \ | _ {l^2（| x | <1）}^2- \ frac {1} {3} {3} \ | u \ | _ {l^3（l^3（|
We study blowup solutions of the 6D energy critical heat equation $u_t=Δu+|u|^{p-1}u$ in $\R^n\times(0,T)$. A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas, Herrero and Velázquez \cite{FilippasHV}. The dimension six is a border case whether a type II blowup can occur or not. Therefore the behavior of the solution is quite different from other cases. In fact, our solution behaves like \[ u(x,t)\approx \begin{cases} λ(t)^{-2}{\sf Q}(λ(t)^{-1}x) & \text{in the inner region: } |x|\simλ(t), -(p-1)^\frac{1}{p-1}(T-t)^{-\frac{1}{p-1}} & \text{in the selfsimilar region: } |x|\sim\sqrt{T-t} \end{cases} \] with $λ(t)=(1+o(1))(T-t)^\frac{5}{4}|\log(T-t)|^{-\frac{15}{8}}$. The local energy $E_\text{loc}(u) =\frac{1}{2}\|\nabla u\|_{L^2(|x|<1)}^2-\frac{1}{3}\|u\|_{L^3(|x|<1)}^3$ of the solution goes to $-\infty$.
