Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring (Q380057)

Consider \(0<r<R\) and set \(\Omega = \{ x\in\mathbb{R}^n\;:\;r < |x| < R \}\), \(n\geq 2\). It is shown that radially symmetric solutions to the Keller-Segel chemotaxis system NEWLINE\[NEWLINE \partial_t u = \mathrm{div} \left( \nabla u - u \nabla v \right)\;, \quad \partial_t v = \Delta v - v + u\;, \quad (x,t)\in\Omega\times (0,\infty)\;, NEWLINE\]NEWLINE supplemented with homogeneous Neumann boundary conditions and non-negative initial conditions \(u_0\in C(\bar{\Omega})\) and \(v_0\in W^{1,\infty}(\Omega)\) exist for all times and are uniformly bounded. This property contrasts markedly with the same problem in a ball for which finite time blowup of solutions takes place, see [\textit{M. Winkler}, J. Math. Pures Appl. 100, No. 5, 748--767 (2013; \url{doi:10.1016/j.matpur.2013.01.020})]. The key argument in the proof is that any radially symmetric \(f\in H^1(\Omega)\) is actually bounded and satisfies NEWLINE\[NEWLINE \|f\|_\infty \leq \varepsilon \|\nabla f\|_2^2 + C(\varepsilon,r,R) (1+\|f\|_1) NEWLINE\]NEWLINE for all \(\varepsilon>0\), the constant \(C(\varepsilon,r,R)\) blowing up as \(r\to 0\).