Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A new result for boundedness of solutions to a quasilinear
  higher-dimensional chemotaxis -- haptotaxis model with nonlinear diffusion   This paper deals with a boundary-value problem for a coupled quasilinear
chemotaxis--haptotaxis model with nonlinear diffusion
$$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla
u)-\chi\nabla\cdot(u\nabla v)-\xi \nabla\cdot(u\nabla w)+\mu u(1-u-w),\\
v_t=\Delta v- v +u,\quad \\ w_t=- vw\\ \end{array}\right. $$ in $N$-dimensional
smoothly bounded domains, where the parameters $\xi ,\chi> 0$, $\mu> 0$. The
diffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u >
0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it is
proved that under the conditions
$$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}
(\chi+\xi\|w_0\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu)_{+}}+1)
(N+\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\|w_0\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}
(\chi+\xi\|w_0\|_{L^\infty(\Omega)})-\mu)_{+}}-1)}{N}}}}},$$ and proper
regularity hypotheses on the initial data, the corresponding initial-boundary
problem possesses at least one global bounded classical solution when $D(0) >
0$ (the case of non-degenerate diffusion), while if, $D(0)\geq 0$ (the case of
possibly degenerate diffusion), the existence of bounded weak solutions for
system is shown. This extends some recent results by several authors.

37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th