On the finite-time blow-up of a non-local parabolic equation describing chemotaxis. (English) Zbl 1212.35233

Summary: The non-local parabolic equation \(v_t=\Delta v+\lambda e^v/\int _\Omega e^v\) in \(\Omega \times (0,T)\) associated with Dirichlet boundary and initial conditions is considered here. This equation is a simplified version of the full chemotaxis system. Let \(\lambda ^*\) be such that the corresponding steady-state problem has no solutions for \(\lambda >\lambda ^*\), then it is expected that blow-up should occur in this case. In fact, for \(\lambda >\lambda ^*\) and any bounded domain \(\Omega \subset \mathbf R^2\) it is proven, using Trudinger-Moser’s inequality, that \(\int _\Omega e^{v(x,t)}dx\to \infty \) as \(t\to T_{max}\leq \infty \). Moreover, in this case, some properties of the blow-up set are provided. For the two-dimensional radially symmetric problem, i.e. when \(\Omega =B(0,1)\), where it is known that \(\lambda ^*=8\pi \), we prove that \(v\) blows up in finite time \(T^*<\infty \) for \(\lambda >8\pi \) and this blow-up occurs only at the origin \(r=0\) (single-point blow-up, mass concentration at the origin).


35K57 Reaction-diffusion equations
35K55 Nonlinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences