Unboundedness for solutions to a degenerate drift-diffusion equation with the L1-supercritical and the energy subcritical exponent

We consider large time behavior of weak solutions to a degenerate drift-diffusion system related to Keller-Segel system with the L1-supercritical and the energy subcritical cases under relaxed weight condition. It is known that the large time behavior of solutions is classified by the invariant norms of initial data. For the L1-critical case, Ogawa-Wakui proved that the negative entropy condition induces the unboundedness of corresponding solutions with the initial data decaying slowly at spacial infinity. Here the result is a continuous analogy of the L1-critical case. Analogous results have been obtained in the theory of nonlinear Schrödinger equations.