论文标题
具有对数相互作用和非局部反应项的凯勒 - 塞格模型
Keller-Segel model with Logarithmic Interaction and nonlocal reaction term
论文作者
论文摘要
我们研究了二维中的keller-segel模型的全局存在和爆炸凯勒 - 塞格模型的全局存在（m_0- \ in {\ r^2} u dx \ right）$。通过从人口的总质量来解决缺乏质量保护的情况下，引入了转型，我们表明，解决方案的定性行为是由关键值$8π$决定增长参数$ M_0 $和初始质量$ M_0 $的决定。 For general solutions, if both $m_0$ and $M_0$ are less than $8π$, solutions exist globally in time using the energy inequality, whereas there are finite time blow-up solutions for $M_0>8π$ (It involves the case $m_0<8π$) with any initial data and $M_0<8π<m_0$ with small initial second moment. We also show the infinite time blow-up for the critical case $M_0=8 π.$ Moreover, in the radial context, we show that if the initial data $u_0(r)<\frac{m_0}{M_0} \frac{8 λ}{(r^2+λ)^2}$ for some $λ>0$, then all the radially symmetric solutions are vanishing in $ l_ {loc}^1（\ r^2）$ as $ t \ to \ infty $。如果初始数据$ u_0（r）> \ frac {m_0} {m_0} \ frac {8λ} {（r^2+λ）^2} $对于某些$λ> 0 $，则可能存在一个径向对称的对称的质量浓度，使得在$ t \ t \ t \ fty $ \ fty上满足质量浓度。
We investigate the global existence and blow-up of solutions to the Keller-Segel model with nonlocal reaction term $u\left(M_0-\int_{\R^2} u dx\right)$ in dimension two. By introducing a transformation in terms of the total mass of the populations to deal with the lack of mass conservation, we exhibit that the qualitative behavior of solutions is decided by a critical value $8π$ for the growth parameter $M_0$ and the initial mass $m_0$. For general solutions, if both $m_0$ and $M_0$ are less than $8π$, solutions exist globally in time using the energy inequality, whereas there are finite time blow-up solutions for $M_0>8π$ (It involves the case $m_0<8π$) with any initial data and $M_0<8π<m_0$ with small initial second moment. We also show the infinite time blow-up for the critical case $M_0=8 π.$ Moreover, in the radial context, we show that if the initial data $u_0(r)<\frac{m_0}{M_0} \frac{8 λ}{(r^2+λ)^2}$ for some $λ>0$, then all the radially symmetric solutions are vanishing in $L_{loc}^1(\R^2)$ as $t \to \infty$. If the initial data $u_0(r)>\frac{m_0}{M_0} \frac{8 λ}{(r^2+λ)^2}$ for some $λ>0$, then there could exist a radially symmetric solution satisfying a mass concentration at the origin as $t \to \infty.$
