Global bounded solution to a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis
The authors study the chemotaxis-convection system \[\begin{cases} u_t = d_1 \Delta u - \nabla \cdot (u \nabla v) + \nabla \cdot (u \nabla w) + f(u), & \; (x,t) \in \Omega \times (0,\infty), \\ v_t = d_2 \Delta v + \nabla \cdot (v \nabla w) + ku - \mu v, & \; (x,t) \in \Omega \times (0,\infty), \\ w_t = d_3 \Delta w + ru - \delta w, & \; (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \((u_0,v_0,w_0)\) with \(u_0 \in W^{1,\infty}(\Omega)\), \(v_0, w_0 \in W^{2,\infty}(\Omega)\) and \(u_0 \not\equiv 0\), where \(\Omega \subset \mathbb{R}^2\) is a bounded domain with smooth boundary. Moreover, \(d_i\), \(k\), \(r\), \(\mu\), \(\delta\) are positive constants and \(f \in C^1([0,\infty))\) satisfies \(f(0) \ge 0\) and \(f(s) \le a - b s^m\) for \(s>0\) with some \(a\ge 0\), \(b>0\), and \(m>3\). The system describes the branching of capillary sprouts during angiogenesis, where \(u\), \(v\), and \(w\) denote the densities of endothelial cells, adhesive sites, and matrix, respectively. Requiring the assumptions given above (in particular \(m>3\)), the authors prove the existence of a unique global and bounded classical solution to the above system such that \(u,v,w \in L^\infty((0,\infty); W^{1,4}(\Omega))\). In the proof the authors mainly establish several a priori estimates which are combined in a clever way to result in the uniform boundedness mentioned above. In particular, results on maximal Sobolev regularity of the Neumann heat semigroup are used to prove some of these estimates.
scientific article; zbMATH DE number 7315349
scientific article