Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system
A simplified version of the Keller-Segel chemotaxis model reads \[ \begin{aligned} \partial_t u = & \nabla\cdot \left( \nabla u - u\;\nabla v \right)\,, \quad (t,x)\in (0,\infty)\times{\mathbb R}^2\,, \\ 0 = & \Delta v - \gamma\;v + u \,, \quad (t,x)\in (0,\infty)\times{\mathbb R}^2\,, \end{aligned} \] where \(u\) and \(v\) denote the concentration of cells and chemoattractant, respectively, and \(\gamma>0\). The local existence and uniqueness of solutions have been proved in previous works for an initial condition \(u_0\) in \(H^s({\mathbb R}^2)\), \(s>1\), and in \(L^q({\mathbb R}^2)\), \(q>1\). A similar result is established here under the only assumption \(u_0\in L^1({\mathbb R}^2)\). In addition, while it was already known that non-negative solutions blow up in finite time if \(\| u_0\| _1>8\pi\) and \(\| | x| ^2 u_0\| _1\) is suitably small (so that \(u_0\) is sufficiently concentrated around \(x=0\)), a more precise bound on \(\| | x| ^2 u_0\| _1\) in terms of \(\| u_0\| _1/(8\pi)\) is given for this blow-up to occur.
scientific article; zbMATH DE number 5298501