Boundedness and stabilization in a haptotaxis model of oncolytic virotherapy with nonlinear sensitivity

This paper deals with a haptotaxis system modeling oncolytic virotherapy with nonlinear sensitivity

$ \begin{eqnarray*} \left\{ \begin{split}{} &u_t = \Delta u - \nabla\cdot(u\chi(v)\nabla v) + \mu u(1-u) - uz, &(x, t)\in \Omega\times (0, \infty), \\ &v_t = -(u+w)v, &(x, t)\in \Omega\times (0, \infty), \\ &w_t = \Delta w - w + uz, &(x, t)\in \Omega\times (0, \infty), \\ &z_t = \Delta z - z - uz +\beta w, &(x, t)\in \Omega\times (0, \infty), \ \end{split} \right. \end{eqnarray*} $

under homogeneous Neumann boundary conditions in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{3} $, where $ \mu, \beta>0 $ and $ \chi(v) $ is a nonlinear sensitivity. Firstly, we obtain the local well-posedness and global boundedness of solutions for the above system. Moreover, when $ \beta\in(0, 1) $, we can prove that the global solution $ (u, v, w, z) $ exponentially stabilizes to the constant equilibrium $ (1, 0, 0, 0) $ in the topology $ L^{p}(\Omega)\times(L^{\infty}(\Omega))^3 $ with $ p>1 $ as $ t \rightarrow \infty $.