Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction
Youshan Tao, Michael Winkler
Abstract
<p style='text-indent:20px;'>We consider the haptotaxis system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which arises as a simplified version of a recently proposed model for oncolytic virotherapy. When posed under no-flux boundary conditions in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document}</tex-math></inline-formula>, with positive parameters <inline-formula><tex-math id="M2">\begin{document}$ D_w $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ D_z $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>, and along with initial conditions involving suitably regular data, this system is known to admit global classical solutions. <p style='text-indent:20px;'>It is shown that with respect to infinite-time blow-up, this system exhibits a critical mass phenomenon related to the quantity <inline-formula><tex-math id="M5">\begin{document}$ m_c: = \frac{1}{(\beta-1)_+} $\end{document}</tex-math></inline-formula>: In fact, it is seen that each solution fulfilling <inline-formula><tex-math id="M6">\begin{document}$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) > m_c $\end{document}</tex-math></inline-formula> must be unbounded, and this is complemented by a boundedness result which inter alia asserts that for any choice of <inline-formula><tex-math id="M7">\begin{document}$ m<m_c $\end{document}</tex-math></inline-formula> one can find a nontrivial set of solutions, particularly containing spatially heterogeneous solutions, each of which is bounded though satisfying <inline-formula><tex-math id="M8">\begin{document}$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) = m $\end{document}</tex-math></inline-formula>.