No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system

We deal with the fully parabolic 1d chemotaxis system with diffusion
$1/(1+u)$. We prove that the above mentioned nonlinearity, despite
being a natural candidate, is not critical. It means that for such a
diffusion any initial condition, independently on the magnitude of
mass, generates global-in-time and bounded solution. In order to prove
our theorem, we establish a new Lyapunov-like functional associated to
the system. This talk is based on a recent joint project with Tomasz
Cieslak.