Some further progress for boundedness of solutions to a quasilinear higher–dimensional chemotaxis–haptotaxis model with nonlinear diffusion

This study focuses on the $ N $-dimensional chemotaxis-haptotaxis model with nonlinear diffusion that was initially proposed by Chaplain and Lolas (see [9]) to describe the interactions between cancer cells, the matrix-degrading enzyme, and the host tissue during cancer cell invasion. Accordingly, we consider the diffusion coefficient $ D(u) $ of cancer cells to be a nonlinear function satisfying $ D(u)\geq C_{D}u^{m-1} $ for all $ u > 0 $ with some $ C_D>0 $ and $ m>0 $. Relying on a new energy inequality and iteration argument, this paper proves that under the mild condition$ m>\frac{2N(N+1)[\max _{2\leq s\leq N+2}\lambda_{0, s}^{\frac{1}{{{s}}}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu]_+}{(N+2) [(N+1)\max _{2\leq s\leq N+2}\lambda_{0, s}^{\frac{1}{{{s}}}} (\chi+\xi\|w_0\|_{L^\infty(\Omega)})-N\mu]_+}, $and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem has at least one globally bounded classical solution when $ D(0) > 0 $ (the case of nondegenerate diffusion), while if $ D(0)\geq 0 $ (the case of possibly degenerate diffusion), the existence of bounded weak solutions for the system is shown, where the positive parameters $ \xi $, $ \chi $, and $ \mu> 0 $ measure the chemotactic and haptotactic sensitivities and proliferation rate of the cells, respectively.