Existence and nonuniqueness of segregated solutions to a class of cross-diffusion systems

Abstract

We study the the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=\operatorname{div}\left(a_iu_i\nabla (u_1+u_2)\right)+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder \(Q=\Omega\times (0,T]\). The functions \(f_i\) are assumed to satisfy the conditions \(f_1(0,r)=0\), \(f_2(s,0)=0\), \(f_1(0,r)\), \(f_2(s,0)\) are locally Lipschitz-continuous. 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. Results of numerical simulation are presented.