Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
The authors study energy solutions to the quasilinear degenerate chemotaxis system \[ \begin{cases} u_t = \Delta u^m - \nabla \cdot (u^{q-1} \nabla v), \quad & (x,t) \in \Omega \times (0,\infty), \\ v_t = \Delta v -v +u, \quad & (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0 \in L^\infty(\Omega)\), \(v_0 \in W^{1,\infty} (\Omega)\) such that \(\nabla u_0^m \in L^2(\Omega)\). Here \(\Omega := B_R(0) \subset \mathbb{R}^N\) is a ball, \(N \geq 2\), \(m \geq 1\), and \(q \geq 2\). It is known that in case of \(q < m + \frac{2}{N}\) any energy solution is global in time and bounded, while for \(q > m + \frac{2}{N}\) there exist blow-up solutions. However, in [\textit{S. Ishida} and \textit{T. Yokota}, Discrete Contin. Dyn. Syst., Ser. B 18, No. 10, 2569--2596 (2013; Zbl 1277.35071)] it remained open whether in the latter case the blow-up occurs in finite or infinite time. The authors now answer this open question and prove that in case of \(q > m + \frac{2}{N}\) for any positive initial mass \(M\) there exists a large class of radially symmetric initial data \((u_0,v_0)\) such that \(\int_\Omega u_0 = M\) and the corresponding energy solution blows up in finite time. As the main ingredient of their proof the author show an appropriate ODI for the Lyapunov functional of the above system. This is done by an extension of the method established by \textit{M. Winkler} [J. Math. Pures Appl. (9) 100, No. 5, 748--767 (2013; Zbl 1326.35053)] and its variant from the quasilinear non-degenerate case from [\textit{T. Cieślak} and \textit{C. Stinner}, J. Differ. Equations 252, No. 10, 5832--5851 (2012; Zbl 1252.35087)] to the quasilinear degenerate case.
quasilinear degenerate Keller-Segel systems
scientific article; zbMATH DE number 6848985
scientific article