Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem

We study the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=div(a_iu_i\nabla (u_1+u_2))+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder $Q=\Omega\times (0,T]$. It is assumed that the functions $f_1(r,0)$, $f_2(0,s)$ are locally Lipschitz-continuous and $f_1(0,s)=0$, $f_2(r,0)=0$. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique and derive the equation of motion of the surface $\Gamma$ which separates the parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $\Gamma$ is a modification of the Darcy law in filtration theory.