论文标题
适应性和爆炸，用于不均匀的半线性抛物线方程
Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation
论文作者
论文摘要
我们考虑不均匀方程的签名解决方案的较大时间行为$ u_t-Δu= | x |^|^|^α| $α> -2 $，$ \ z，{\ mathbf w} $是连续的函数，因此$ζ（t）= t^σ$或$ζ（t）\ sim t^σ$ as $ t \ to $ t \ to 0 $，$ζ（t）\ sim t^m $ as $ t \ to \ t \ to \ infty $。我们以$σ> -1 $获得本地存在。我们还显示以下内容：\ begin {inatizize} \ item如果$ m \ leq 0 $，$ p <\ frac {n-2m+α} {n-2m-2} $和$ \ int {\ int {\ intbb {r}^n}^n}^n} \ item如果$ m> 0 $，$ p> 1 $和$ \ int _ {\ mathbb {r}^n} {\ mathbf w}（x）dx> 0 $，则所有解决方案在有限的时间内爆炸； \ item如果$ζ（t）= t^σ$带有$ -1 <σ<0 $，则为$ u_0：= u（t = 0）$和$ {\ mathbf w} $足够小，解决方案在全球范围内存在。 \ end {inatize}我们还讨论了较低的维度。本文的主要新颖性是，爆炸取决于无限$ζ$的行为。
We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-Δu=|x|^α|u|^{p}+ζ(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $α>-2$, $\z, {\mathbf w}$ are continuous functions such that $ζ(t)=t^σ$ or $ζ(t)\sim t^σ$ as $t\to 0$, $ζ(t)\sim t^m$ as $t\to\infty$ . We obtain local existence for $σ>-1$. We also show the following: \begin{itemize} \item If $m\leq 0$, $p<\frac{N-2m+α}{N-2m-2}$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $m> 0$, $p>1$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $ζ(t)=t^σ$ with $-1<σ<0$, then for $u_0:=u(t=0)$ and ${\mathbf w}$ sufficiently small the solution exists globally. \end{itemize} We also discuss lower dimensions. The main novelty in this paper is that blow up depends on the behavior of $ζ$ at infinity.
