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A fast blowup solution to an elliptic-parabolic system related to chemotaxis. (English) Zbl 1154.35058
The parabolic-elliptic system
\[ \begin{aligned} \partial_t u = \nabla\cdot \left( \nabla u - u\;\nabla v \right), &\quad (t,x)\in (0,T)\times\mathbb{R}^N,\\ 0=\Delta v + u, &\quad (t,x)\in (0,T)\times\mathbb{R}^N, \end{aligned} \] has positive and radially symmetric blowing-up (or backward) self-similar solutions of the form \(u(t,x)=(T-t)^{-1} U(x(T-t)^{-1/2})\) for their first component when \(N\geq 3\). The existence of a positive and radially symmetric solution blowing up at a faster rate is established herein if \(N\geq 11\). A lower bound of the \(L^\infty\)-norm of the first component of this solution near the blow-up time is actually provided and its behaviour is described in a suitable region of the time-space variables. As the blow-up occurs at a faster rate than the one given by the previously known self-similar solution, this phenomenon is refered to as Type-II blow-up. The proof uses an argument similar to that of the same result for \(\partial_t w = \Delta w + w^p\) for \(N\geq 11\) in \((0,T)\times\mathbb{R}^N\). A more precise result concerning the blow-up rate has been obtained subsequently in [N. Mizoguchi and T. Senba, Adv. Math. Sci. Appl. 17, No. 2, 505–545 (2007; Zbl 1147.35013)].

35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K45 Initial value problems for second-order parabolic systems
92C17 Cell movement (chemotaxis, etc.)