论文标题
反应的不均匀多孔培养基方程的爆炸和全球存在
Blow-up and global existence for the inhomogeneous porous medium equation with reaction
论文作者
论文摘要
我们研究了有限的时间爆破和全球存在解决方案的解决方案，用于cauchy问题的多孔培养基方程，其密度$ρ（x）$和类似功率的反应项。我们表明，对于足够小的初始数据，如果$ρ（x）\ sim \ frac {1} {\ left（\ log | x | \ right）^α| x | x |^{2}} $ as $ | x | \ to \ infty $，那么任何$ p> 1 $的解决方案都存在于全球范围内。 On the other hand, when $ρ(x)\sim\frac{\left(\log|x|\right)^α}{|x|^{2}}$ as $|x|\to \infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $ p> m $。这些结果概括了[27]和[28]中建立的结果，其中$ρ（x）\ sim | x | x |^{ - q} $ for $ q> 0 $ as as $ | x | \ to \ to \ infty $。
We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term. We show that for small enough initial data, if $ρ(x)\sim \frac{1}{\left(\log|x|\right)^α|x|^{2}}$ as $|x|\to \infty$, then solutions globally exist for any $p>1$. On the other hand, when $ρ(x)\sim\frac{\left(\log|x|\right)^α}{|x|^{2}}$ as $|x|\to \infty$, if the initial datum is small enough then one has global existence of the solution for any $p>m$, while if the initial datum is large enough then the blow-up of the solutions occurs for any $p>m$. Such results generalize those established in [27] and [28], where it is supposed that $ρ(x)\sim |x|^{-q}$ for $q>0$ as $|x|\to \infty$.
