Large time behavior and uniqueness of solutions of weakly coupled system of reaction-diffusion equations (Q1429086)

The paper considers nonnegative solutions of the initial value problem for a weakly coupled system \[ \frac{\partial u_i(x,t)}{\partial t} = \Delta u_i(x,t)+ u_{i+1}^p(x,t) \] with a nonnegative initial condition and \(x\in\mathbb{R}^d\), \(t>0\) and \(i=1,\dots ,n\). It complements the results of \textit{N. Umeda} [Tsukuba J. Math. 27, 31--46 (2003, Zbl 1035.35018)] by considering \(\prod_{i=1}^n p_i \leq 1\) with \(p_i >0\). It proves global existence of solutions of such a system and provides conditions for the uniqueness and asymptotic behaviour of the solutions. For \(0<\prod_{i=1}^n p_i <1\) and trivial initial condition, nontrivial solutions can be expressed in a closed form.