Boundedness in a chemo-repulsion model with nonlinear sensitivity and logistic source

This paper investigates the Neumann initial-boundary value problem of the chemo-repulsion model: $ u_t = \Delta u+ \nabla\cdot(u\chi(v)\nabla v)+au-bu^2,\:v_t = \Delta v-v+u $, where $ a,b>0 $ and $ \chi(\cdot)\in C^2([0,\infty)) $ satisfies $ 0\le \chi(s)\le \chi_0(1+s)^k $ for all $ s\ge0 $, with some $ \chi_0>0 $. By constructing a family of functionals: $ \mathcal{E}_p(t): = \int_{\Omega} uv^{2p} +\int_{\Omega} |\nabla v^{p+1}|^2 $ for $ p\ge1 $, the global boundedness of classical solutions is proved for all $ k\ge0 $ in two-dimensional.