Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction

We consider the haptotaxis system

$ \begin{eqnarray*} \left\{ \begin{array}{lcl} u_t & = & \Delta u - \nabla \cdot (u\nabla v), \\ v_t & = & - (u+w)v, \\ w_t & = & D_w \Delta w - w + uz, \\ z_t & = & D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*} $

which arises as a simplified version of a recently proposed model for oncolytic virotherapy. When posed under no-flux boundary conditions in a smoothly bounded domain $ \Omega\subset \mathbb{R}^2 $, with positive parameters $ D_w $, $ D_z $ and $ \beta $, and along with initial conditions involving suitably regular data, this system is known to admit global classical solutions.

It is shown that with respect to infinite-time blow-up, this system exhibits a critical mass phenomenon related to the quantity $ m_c: = \frac{1}{(\beta-1)_+} $: In fact, it is seen that each solution fulfilling $ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) > m_c $ must be unbounded, and this is complemented by a boundedness result which inter alia asserts that for any choice of $ m<m_c $ one can find a nontrivial set of solutions, particularly containing spatially heterogeneous solutions, each of which is bounded though satisfying $ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) = m $.