Convergence of solutions of reaction-diffusion systems with time delays.

*(English)*Zbl 0992.35105The paper deals with a class of reaction-diffusion-convection systems with time delays in a bounded domain \(\Omega\in {\mathbb R^p}\) under Neumann boundary condition
\[
\begin{aligned} \partial u_i/\partial t-L_i u_i=f_i({\mathbf u}, J\ast {\mathbf u}), &\quad t>0,\;x\in \Omega,\\ \partial u_i/\partial\nu=0, &\quad t>0,\;x\in \partial\Omega,\\ u_i(t,x)=\eta_i(t,x) &\quad t\in I_i,\;x\in \Omega,\quad i=1,\dots,N.\end{aligned}
\]
The author studies the asymptotic behavior of the time-dependent solution in relation to constant steady-state solutions, including regions of attraction of the stable steady solutions. The obtained results are applied to two Volterra-Lotka models in ecology for studying the global stability and instability of the various constant steady-state solutions. The stability conditions for these model problems are given in terms of the rate constants of the reaction and are independent of the time delays and the effect of diffusion.

Reviewer: Dian K.Palagachev (Bari)

##### Keywords:

stability conditions; asymptotic behavior; global attractor; upper and lower solutions; Volterra-Lotka models in ecology
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\textit{C. V. Pao}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 48, No. 3, 349--362 (2002; Zbl 0992.35105)

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