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

Abstract

We consider the parabolic chemotaxis model \[ u_t=Δu - χ\nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=Δv - v + u\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $χ\in(0,χ_0)$ for some $χ_0>1$, thereby proving that the value $χ=1$ is not critical in this regard. Our main tool is consideration of the energy functional \[ \mathcal{F}_{a,b}(u,v)=\int_Ωu\ln u - a \int_Ωu\ln v + b \int_Ω|\nabla \sqrt{v}|^2 \] for $a>0$, $b\geq 0$, where using nonzero values of $b$ appears to be new in this context.