| 杨春晓,杨金戈,郑斯宁.具位势项的快扩散方程的第二临界指标[J].数学研究及应用,2012,32(6):715~722 |
| 具位势项的快扩散方程的第二临界指标 |
| The Second Critical Exponent for a Fast Diffusion Equation with Potential |
| 投稿时间：2012-02-19 修订日期：2012-03-27 |
| DOI：10.3770/j.issn:2095-2651.2012.06.011 |
| 中文关键词: 第二临界指标 快扩散方程 位势 整体解 爆破 |
| 英文关键词:the second critical exponent fast diffusion equation potential global solutions blow-up. |
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| 中文摘要: |
| 本文考虑带有位势项的快扩散方程~$u_t=\Delta u^m-V(x)u^m+u^p$, $(x,t)\in\mathbb{R}^n\times(0,T)$,其中 ~$1-\frac{2}{\alpha m+n}1$, $n\geq 2$, ~$V(x)\sim\frac{\omega}{\mid x\mid^2}$当 ~$|x|\to \infty$, ~$\omega\geq 0$,~$\alpha$ 为 ~$\alpha m(\alpha m+n-2)-\omega=0$ 的正解.我们已在先前的文章得到该方程的 ~Fujita 临界指标~$p_c=m+\frac{2}{\alpha m+n}$. 本文继续建立其第二临界指标, 即在整体解与非整体解共存的参数区域~$p>p_c$,通过初值在无穷远处的衰减阶进一步区分解的整体存在与有限时刻爆破.我们证明: 若~$u_0(x)\sim |x|^{-a}$当 ~$|x|\rightarrow \infty$,则第二临界指标 ~$a^*=\frac{2}{p-m}$. 与 ~Fujita 临界指标 ~$p_c$不同的是,这里得到的第二临界指标与位势系数 ~$\omega$ 无关. |
| 英文摘要: |
| This paper considers a fast diffusion equation with potential $u_t=\Delta u^m-V(x)u^m+u^p$ in $\mathbb{R}^n\times(0,T)$, where $1-\frac{2}{\alpha m+n}1$, $n\geq 2$, $V(x)\sim \frac{\omega}{\mid x\mid^2}$ with $\omega\geq 0$ as $|x|\rightarrow \infty$, and $\alpha$ is the positive root of $\alpha m(\alpha m+n-2)-\omega=0$. The critical Fujita exponent was determined as $p_c=m+\frac{2}{\alpha m+n}$ in a previous paper of the authors. In the present paper, we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region $p>p_c$ via the critical decay rates of the initial data. With $u_0(x)\sim |x|^{-a}$ as $|x|\rightarrow \infty$, it is shown that the second critical exponent $a^*=\frac{2}{p-m}$, independent of the potential parameter $\omega$, is quite different from the situation for the critical exponent $p_c$. |
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