论文标题
具有持续奇异性的非线性热方程的签名解决方案
Sign-changing solutions of the nonlinear heat equation with persistent singularities
论文作者
论文摘要
我们研究了非线性热方程$ \ partial _t u =ΔU + | u | u |^αu$上的签名改变解决方案的存在， {n-4+2 \ sqrt {n-1}} \ in（\ frac {2} {n-2} {n-2}，\ frac {4} {n-2}）$，在时间间隔的时间为$ x = 0 $。特别是，对于某些可以任意大的$μ> 0 $，我们证明，对于任何$ u_0 \ in \ mathrm {l}^\ infty _ {\ mathrm {\ mathrm {loc}}}（{\ mathbb r} {2} {α}} $在$ 0 $的社区中，存在局部（及时）的（及时）$ u $的非线性加热方程，具有初始值$ u_0 $，它是签名更换的，在无限范围内，在无限范围内，具有单界$β| | | | x | x | x | x | x |^{ - \ frac { - \ frac {2} $ $ | x |^{\ frac {2} {α}} u（t，x）\toβ$ as $ | x | \至0 $，其中$β= \ frac {2} {α}（n -2 - \ frac {2} {α}）$。通常，这些解决方案既不是静止的也不相似。
We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = Δu + |u|^αu$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < α<α_0$, where $α_0=\frac {4} {N-4+2\sqrt{ N-1 } }\in (\frac {2} {N-2}, \frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $μ>0$ that can be arbitrarily large, we prove that for any $u_0 \in \mathrm{L} ^\infty _{\mathrm{loc}} ({\mathbb R}^N \setminus \{ 0 \}) $ which is bounded at infinity and equals $μ|x|^{- \frac {2} {α}}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $β|x|^{- \frac {2} {α}}$ at the origin in the sense that for $t>0$, $ |x|^{\frac {2} {α}} u(t,x) \to β$ as $ |x| \to 0$, where $β= \frac {2} {α} ( N -2 - \frac {2} {α} ) $. These solutions in general are neither stationary nor self-similar.
