A quasilinear chemotaxis-haptotaxis system: existence and blow-up results (Q6584922)

Let \(\Omega\) be a bounded domain of \(\mathbb{R}^n\), \(n\ge 3\), with smooth boundary and consider the chemotaxis-haptotaxis system\N\begin{align*}\N\partial_t u & = \mathrm{div} \big( D(u)\nabla u - \chi S(u) \nabla v - \xi H(u) \nabla w \big)\quad & \text{in } (0,\infty)\times\Omega, \\\N\partial_t v & = \Delta v -v + u \quad & \text{in } (0,\infty)\times\Omega, \\\N\partial_t w & = - v w \quad & \text{in }(0,\infty)\times\Omega,\N\end{align*}\Nsupplemented with no-flux boundary conditions and sufficiently smooth non-negative initial conditions, where \(\chi\) and \(\xi\) are two positive real numbers.\N\NIf \(H=\mathrm{id}\), \(D\in C^2([0,\infty))\), and \(S\in C^2([0,\infty))\) are such that \(S(0)=0\),\N\begin{align*}\NS(s) \le A (s+1)^\alpha D(s), \quad A_0(s+1)^{m-1} \le D(s) \le A_1 (s+1)^{M-1}, \qquad s\ge 0,\N\end{align*}\Nwith \(\alpha<2/n\), \(M\ge m>2(n-1)/n\), and \((A,A_0,A_1)\in (0,\infty)^3\), then the existence and uniqueness of a bounded global classical solution are shown.\N\NIf \(H=S\) and \(S(s)\ge K_1 s^{2/n} D(s)\) for \(s\ge 1\), along with an additional technical assumption on \(D/S\), then there are solutions which blow up in finite or infinite time.\N\NFinally, if \(D=S\equiv 1\) and \(H=\mathrm{id}\), then global bounded solutions are shown to exist for suitably small initial data.