论文标题
爆破以进行完全小数的热方程
Blow-up for a fully fractional heat equation
论文作者
论文摘要
我们研究对完全分数热方程$$ \ MATHCAL {m} u = u^p，\ qquad x \ in \ Mathbb {r}^n，\; 0 <t <t $ 4 with $ p> 0 $ a $ \ mathcal {m} $ a a a a a a a a a a a a a a a a a a i a a a a i a a a a i a a i a a a a in a a a ivane a i a n a n a n n local操作员$ m（x，t）= c_ {n，σ} t^{ - \ frac n2-1-σ} e^{ - \ frac {| x |^2} {4t}} {1} {1} _ {\ {\ {t> 0 \}} $，$ 0 <σ<1 $ $。该操作员与热运算符的分数功率相吻合，$ \ MATHCAL {M} =（\ partial_t-δ）^σ$通过Semigroup理论定义。我们表征了全局存在指数$ p_0 = 1 $和fujita指数$ p _*= 1+ \ frac {2σ} {n+2（1-σ）} $，并研究低于$ p _*$倾向于Infinity的爆炸溶液的速率， （t-t）^{ - \fracσ{p-1}} $。
We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation $$ \mathcal{M} u=u^p,\qquad x\in\mathbb{R}^N,\;0<t<T $$ with $p>0$, where $\mathcal{M}$ is a nonlocal operator given by a space-time kernel $M(x,t)=c_{N,σ}t^{-\frac N2-1-σ}e^{-\frac{|x|^2}{4t}}{1}_{\{t>0\}}$, $0<σ<1$. This operator coincides with the fractional power of the heat operator, $\mathcal{M}=(\partial_t-Δ)^σ$ defined through semigroup theory. We characterize the global existence exponent $p_0=1$ and the Fujita exponent $p_*=1+\frac{2σ}{N+2(1-σ)}$, and study the rate at which the blowing-up solutions below $p_*$ tend to infinity, $\|u(\cdot,t)\|_\infty\sim (T-t)^{-\fracσ{p-1}}$.
