A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity

We consider the parabolic chemotaxis model \[ u_t=\Delta u - \chi \nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=\Delta v - v + u\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $\chi\in(0,\chi_0)$ for some $\chi_0>1$, thereby proving that the value $\chi=1$ is not critical in this regard. Our main tool is consideration of the energy functional \[ \mathcal{F}_{a,b}(u,v)=\int_\Omega u\ln u - a \int_\Omega u\ln v + b \int_\Omega |\nabla \sqrt{v}|^2 \] for $a>0$, $b\geq 0$, where using nonzero values of $b$ appears to be new in this context.