Global asymptotics in some quasimonotone reaction-diffusion systems with delays
The authors consider the asymptotic behaviour for positive solutions of quasimonotone delayed reaction-diffusion systems on bounded domains \(\Omega\subset \mathbb{R}^N\). To be more specific: \(\mathbb{R}^k\)-valued solutions \(u(t,x)\) satisfy the system \[ {\partial\over\partial t} u_i(x,t)= D_i\Delta u_i(x,t)+ F_i(x, u_{t,1}(x),\dots, u_{t,k}(x)),\quad t>0,\quad x\in\Omega, \] \[ {\partial\over\partial\nu} u_i(x,t)+ \alpha_i(x)u_i(x, t)=0,\quad t>0,\quad x\in\partial\Omega, \] \[ u_i(x,t)= \phi_i(x,t)\geq 0,\quad t\in[-\tau_i, 0],\quad x\in\Omega, \] where \(u_{t,i}(x)= u_i(x,\cdot): [t-\tau_i, t]\to\mathbb{R}\), the restriction to the last time interval of length \(\tau_i\) before \(t\). Next to the condition that \(F\) is quasimonotone, their basic assumptions are that \(F\) is: 1) positive: \(F(x,\phi)\geq 0\), 2) subhomogeneous: \(F(x,\alpha\phi)\geq \alpha F(x,\phi)\) and similar for a derivative, and 3) irreducible (also for a derivative). Depending on the sign of \(\lambda_0\), which is the first eigenvalue of the corresponding stationary linearized problem, they show (let me skip the lengthy technical conditions and just give a rough description) that i) if \(\lambda_0\leq 0\), then \(u=0\) is a global attractor; ii) if \(\lambda_0>0\), then either there exists a positive steady state which is a global attractor for initial values \(0\neq\phi\geq 0\), or solutions become arbitrarily large as \(t\to\infty\) for initial values \(0\neq \phi\geq 0\).
scientific article; zbMATH DE number 1038938
scientific article