Boundedness in a quasilinear chemotaxis–haptotaxis system with logistic source

Abstract

In this paper, we consider the quasilinear chemotaxis–haptotaxis system

$$\begin{aligned}\left\{\begin{array}{ll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S_1(u)\nabla v)-\nabla\cdot(S_2(u)\nabla w)+uf(u,w),\quad x\in\Omega,~t > 0,v_t=\Delta v-v+u,\quad x\in\Omega,~t > 0,w_t=-vw,\quad x\in\Omega,~t > 0\end{array} \right.\end{aligned}$$

(⋆)

in a bounded smooth domain ${\Omega\subset\mathbb{R}^n~(n\geq1)}$ under zero-flux boundary conditions, where the nonlinearities ${D,~S_1}$ and ${S_2}$ are assumed to generalize the prototypes

$$D(u)=C_{D}(u+1)^{m-1},~S_1(u)=C_{S_1}u(u+1)^{q_1-1} \quad {\mathrm{and}} \quad S_2(u)=C_{S_2}u(u+1)^{q_2-1}$$

with ${C_D,C_{S_1},C_{S_2} > 0,~m,q_1,q_2\in\mathbb{R}}$ and ${f(u,w)\in C^1([0,+\infty)\times[0,+\infty))}$ fulfills

$$f(u,w)\leq r-bu\quad {\mathrm{for all}}~~u\geq 0\quad {\mathrm{and}} \quad w\geq 0,$$

where ${r > 0,~b > 0.}$ Assuming nonnegative initial data ${u_0(x)\in W^{1,\infty}(\Omega),v_0(x)\in W^{1,\infty}(\Omega)}$ and ${w_0(x)\in C^{2,\alpha}(\bar\Omega)}$ for some ${\alpha\in(0,1),}$ we prove that (i) for ${n\leq2,}$ if ${\max\{q_1,q_2\} < m+\frac{2}{n}-1,}$ then ${(\star)}$ has a unique nonnegative classical solution which is globally bounded, (ii) for ${n > 2,}$ if ${\max\{q_1,q_2\} < m+\frac{2}{n}-1}$ and ${m > 2-\frac{2}{n}}$ or ${\max\{q_1,q_2\} < m+\frac{2}{n}-1}$ and ${m\leq 1,}$ then ${(\star)}$ has a unique nonnegative classical solution which is globally bounded.