Title: Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity,
Abstract: The paper should be viewed as complement of an earlier result in [8]. In the
paper just mentioned it is shown that 1d case of a quasilinear
parabolic-elliptic Keller-Segel system is very special. Namely, unlike in
higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear
diffusion of the form 1/u all the solutions, independently on the magnitude of
initial mass, stay bounded. However, the argument presented in [8] deals with
the Jager-Luckhaus type system. And is very sensitive to this restriction.
Namely, the change of variables introduced in [8], being a main step of the
method, works only for the Jager-Luckhaus modification. It does not seem to be
applicable in the usual version of the parabolic-elliptic Keller-Segel system.
The present paper fulfils this gap and deals with the case of the usual
parabolic-elliptic version. To handle it we establish a new Lyapunov-like
functional (it is related to what was done in [8]), which leads to global
existence of the initial-boundary value problem for any initial mass.