اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدInfinite time blow-up in the Patlak-Keller-Segel system: existence and stability
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Perhaps the most classical diffusion model for chemotaxis is the Patlak-Keller-Segel system begin{equation} label{ks0} left{ begin{aligned} u_t =& Delta u - abla cdot(u abla v) quad inn R^2times(0,infty), v =& (-Delta_{R^2})^{-1} u := frac 1{2pi} int_{R^2} , log frac 1{|x-z|} u(z,t), dz, & qquad u(cdot ,0) = u_0 ge 0quadhbox{in } R^2. end{aligned} right. end{equation} We consider the {em critical mass case} $int_{R^2} u_0(x), dx = 8pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of equ{ks0} is globally defined and blows-up in infinite time. As $tto+infty $ it has the approximate profile $$ u(x,t) approx frac 1{la^2} U_0left (frac {x-xi(t)}{la(t)} right ), quad U_0(y)= frac{8}{(1+|y|^2)^2}, $$ where $la(t) approx frac c{sqrt{log t}}, xi(t)to q $ for some $c>0$ and $qin R^2$
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