Spatial segregation limit of a non-autonomous competition-diffusion system (Q663689)

Let \(\Omega \subset \mathbb{R}^n\) be a bounded domain, \(Q_T:=\Omega\times(0,T)\). The following problem with unknown functions \(u(x,t)\) and \(v(x,t)\) is considered: \[ \begin{aligned} u_t-\Delta u &= f(u)- \alpha_1(x)\,k\,u\,v \text{ in } Q_T, \\ v_t-\Delta v &= g(v)- \alpha_2(x)\,k\, u\, v \text{ in } Q_T;\qquad u=0,\;v=0 \text{ on } \partial \Omega \times (0,T); \\ u(x,0)&=u_0(x), \\ v(x,0)&=v_0(x) \text{ in } \Omega \end{aligned} \] where \(k\) is a large parameter, \(u_0, v_0 \in C^2 (\overline{\Omega})\), \ \(0\leq u_0, v_0\leq M_0\), \ \(u_0\,v_0 = 0\) a.e. in \(\Omega\); \ \(f(z)>0,\;g(z)>0\) for \(z\in (0,M_0)\) and \(f(z)<0,\;g(z)<0 \) for \(z>M_0\). This problem arises in population ecology. The authors study the convergence of the nonnegative solution \(u^k(x,t), \;v^k(x,t)\) of this problem as \(k\to\infty\). They prove that there exist the functions \(u(x,t), v(x,t)\) such that up to the subsequences \(u^k \to u\), \(v^k \to v\), \(w^k:= \alpha_2 u^k -\alpha_1 v^k \to w\) in \(L^2(0,T;\,H_0^1(\Omega))\) as \(k\to\infty\) and \(u\,v = 0\) a.e. in \(Q_T\), where \(w\) is the solution of an initial-boundary value problem given in the paper. If \(\alpha_1(x)=\alpha_2(x)\), then \(u^k \to u\), \(v^k \to v\) uniformly in \(C(\overline{Q}_T)\) as \(k\to\infty\).