具有弱超线性吸收的扩散方程的可允许初始增长性
Admissible initial growth for diffusion equations with weakly superlinear absorption
摘要：
研究了当$f(u)$是一个连续函数，{ \it mildly}在无穷远处超线性时，模型情形为$f(u)=u\ln^\ga (1+u)$与$1\textless{}\ga\textless{}2$的正解在$\BBR\_+\ti\BBR^N$中的初始值在无穷远处的容许增长性.我们特别证明了，如果初始数据在无穷远处的增长太强，则不再有扩散，相应的解满足$\BBR\_+$上的常微分方程问题$\prt\_t \gf+f(\gf)=0$，其中有$\gf(0)=\infty$ .
英文摘要：
We study the admissible growth at infinity of initial data of positive solutions of $\prt\_t u-\Gd u+f(u)=0$ in $\BBR\_+\ti\BBR^N$ when $f(u)$ is a continuous function, {\it mildly} superlinear at infinity, the model case being $f(u)=u\ln^\ga (1+u)$ with $1\textless{}\ga\textless{}2$. We prove in particular that if the growth of the initial data at infinity is too strong, there is no more diffusion and the corresponding solution satisfies the ODE problem $\prt\_t \gf+f(\gf)=0$ on $\BBR\_+$ with $\gf(0)=\infty$.
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