论文标题
半线性热方程的全球存在和渐近行为
Global existences and asymptotic behavior for semilinear heat equation
论文作者
论文摘要
在本文中，我们考虑了$ l^2 $至关重要的半线性热方程$的全球库奇问题$ \ partial_t h =ΔH\ pm | h |^{\ frac4d} h，$ 使用$ h（0，x）= h_0 $，其中$ h $是在$ \ r^+\ times \ r^d $上定义的未知实际功能。在大多数有关此主题的研究中，初始数据$ h_0 $属于某些$ p \ ge 2 $的lebesgue Spaces $ l^p（\ r^d）$，或以$ s> 0 $ $ s> 0 $。 {\ it首先，}我们证明，根据$ d $的不同，存在一些正常数$γ_0$，因此，对于任何初始数据$ H_0 $，Cauchy问题在本地和全球范围内都有良好的范围，该数据是径向的，从起源和负sobolev space $ \ dot $ \ dot $ \ dot h^^{ - γ_0} $ \ r^dot $ \ dot $ \ dot。特别是，它导致了上面在$ l^p（\ r^d）$的适当子空间中考虑的cauchy问题的本地和全球解决方案，并带有一些$ p <2 $。 {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$ are obtained with arbitrary large initial data $h_0\in \dot H^{-γ_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$.
In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=Δh\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. {\it First,} we prove that there exists some positive constant $γ_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from the origin and in the negative Sobolev space $\dot H^{-γ_0}(\R^d)$. In particular, it leads to local and global existences of the solutions to Cauchy problem considered above for the initial data in a proper subspace of $L^p(\R^d)$ with some $p<2$. {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$ are obtained with arbitrary large initial data $h_0\in \dot H^{-γ_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$.
