自相似解的扰动
Perturbations of self-similar solutions
摘要：
本文研究了具有$α>0$的非线性热方程$u_t = Δu + |u|^αu$，无论是在${\mathbb R}^N $，$N\ge 1$上，还是在具有Dirichlet边界条件的有界区域上.证明了在Sobolev次临界情形$(N-2) α<4$中，对于每个$μ\in {\mathbb R}$，如果初始值$u_0$在某个$x_0\in Ω$的邻域内满足$u_0 (x) = μ|x-x_0|^{-\frac {2} {α}}$且在该邻域外有界，则热方程以$u(0)= u_0$为初始条件存在无穷多个解.该证明使用不动点参数来构造初始值为$μ|x-x_0|^{-\frac {2} {α}}$的自相似解在${\mathbb R}^N $上的扰动。此外，若$μ\ge μ_0$对某个$ μ_0( N, α)\ge 0$，且$u_0 I\ge 0$，则热方程在初始条件$u(0)= u_0$下不存在非负局部解，但存在无穷多个变号解.
英文摘要：
We consider the nonlinear heat equation $u_t = Δu + |u|^αu$ with $α>0$, either on ${\mathbb R}^N $, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) α<4$, for every $μ\in {\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = μ|x-x_0|^{-\frac {2} {α}}$ in a neighborhood of some $x_0\in Ω$ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $μ|x-x_0|^{-\frac {2} {α}}$ on ${\mathbb R}^N $. Moreover, if $μ\ge μ_0$ for a certain $ μ_0( N, α)\ge 0$, and $u_0 I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
