Global existence and large time behavior of solutions to a haptotaxis model with self-remodeling mechanisms. (Chinese. English summary) Zbl 1499.35432
Summary: In this paper, we deal with the initial-boundary value problem for the following haptotaxis model with self-remodeling mechanisms: \[\begin{cases}u_t=\Delta u-\xi\nabla\cdot(u\nabla w),\\ u_t=\Delta v-v+u,\\w_t=-vw+\eta w(1-u-w)\end{cases}\] in a bounded domain of \(\mathbb{R}^2\) with zero-flux boundary conditions. We show that for any \(\eta > 0\), there exists a unique global classical solution. In particular, we show that the solution is uniformly bounded when \(\eta\) is appropriately small. On the basis of this, we also establish the global asymptotic stability of the constant steady state \((\overline{u}_0, \overline{u}_0, 0)\).
MSC:
35M10 | PDEs of mixed type |
35A09 | Classical solutions to PDEs |
35B35 | Stability in context of PDEs |
92C17 | Cell movement (chemotaxis, etc.) |