Global attractivity and singular perturbation for a class of nonlinear cooperative systems
This fine work deals with the parabolic system \[ \begin{cases} u_t- d_1\Delta u= au+ bv- f(u)\\v_t-d_2\Delta v= cu+ dv- g(v)\end{cases}\quad\text{in }\Omega\times (0,T)\tag{1} \] (\(\Omega\subset\mathbb{R}^N\) bounded) with initial values \((u_0,v_0)\) at \(t=0\) and zero boundary values, where \(d_1\), \(d_2\), \(b\), \(c>0\) and \(f(0)= f'(0)= 0\), \(f(s)/s\to\infty\), \(f'(s)\to\infty\) as \(s\to\infty\), and the same conditions for \(g\). The corresponding steady state problem for \((u(x),v(x))\) (with \(u=v=0\) on \(\partial\Omega\)) and purely kinetic problem for \((u(t),v(t))\) (with \(u(0)\), \(v(0)\) given) are denoted by (2) and (3), respectively. The asymptotic behavior of system (1) depends on \(d_1\), \(d_2\) as follows: The first quadrant in a \((d_1,d_2)\)-plane is divided by a hyperbola \(H\) in two sets \(H_u\) (above \(H\)) and \(H_d\). System (2) has a positive solution (coexistence state) if and only if \((d_1,d_2)\in H_d\), and it is unique and a global attractor for (1), while for \((d_1,d_2)\in H_u\) the zero solution is a global attractor (for positive \(u_0\), \(v_0\)). System (3) has a unique constant positive solution (equilibrium), which is globally asymptotically stable. The coexistence state of (2) converges to this equilibrium as \(d_1, d_2\to 0\), uniformly in compact subsets of \(\Omega\). A main tool is a strong maximum principle, which applies to problem (2).
scientific article; zbMATH DE number 909996