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

TL;DR

This paper proves global existence and boundedness of solutions in a 2D chemotaxis system with singular sensitivity for certain parameter ranges, showing that the critical value is not a limiting factor.

Contribution

It introduces a novel energy functional with a nonzero gradient term for v, extending the understanding of boundedness in chemotaxis models.

Findings

Solutions are globally bounded for h0>1

The critical value h0=1 is not limiting for boundedness

The new energy functional approach is effective in this context

Abstract

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.