| Abstract: |
| We discuss the following quasilinear chemotaxis-haptotaxis system:
$\begin{align*}
u_{t} &= \nabla\cdot(D(u)\nabla u)- \chi \nabla\cdot(S(u)\nabla v)-\xi\nabla\cdot(u\nabla w), \; x \in \Omega\text{, }t>0, \
v_{t} &=\Delta v -v+ u, \; x \in \Omega\text{, }t>0, \
w_{t} &=-vw, \; x \in \Omega\text{, }t>0, \
\end{align*}$
under homogeneous Neumann boundary conditions in a smooth, bounded domain $\Omega \subset \mathbb{R}^{n}, n\geq 3.$ We show that for $\frac{S(s)}{D(s)} \leq A (s+1)^{\alpha}$ for $\alpha < \frac{2}{n}$ and under suitable growth conditions on $D,$ there exists a uniform-in-time bounded classical solution to the above system. Also, we establish that for radial domains, when the opposite inequality is satisfied, the corresponding solutions blow-up in finite or infinite-time. |
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