On a time-dependent bio-reactor model with chemotaxis. (Q1855808)

Let \[ A_i(t)u=-a_i(x,t)u_{xx}+b_i(x,t)u_x+c_i(x,t)u,\quad i=0,1,\cdots,m. \] The author considers the following parabolic system \[ \begin{cases} \frac{\partial{u_0}}{\partial t}+A_0(t)u_0=f_0(x,t,\vec{u}),&x\in \Omega=(0,1),\\ \frac{\partial{u_i}}{\partial t}+A_i(t)u_i+(u_i\Phi_i(u_0)(u_0)_x)_x=f_i(x,t,\vec{u}),&x\in \Omega,\\ B_0(t)u_0(x,t)=S(x,t),\;B_i(t)u_i(x,t)=0,&x\in \partial\Omega=\{0,1\},\\ u_i(x,0)=u^i_0(x),\;x\in \Omega,&\vec {u}=(u_0,u_1,\cdots,u_m). \end{cases} \tag{1} \] (1) is the general form of many systems discussed in the references. By showing that solutions of (1) are ultimately uniformly bounded and that the corresponding dynamical system (autonomous system) of (1) possesses a global attractor in the space \(X=\prod_0^mH^1(0,1)\), the global existence result of \textit{X. Wang} [SIAM Math. Anal. 31, 535--560 (2000; Zbl 0990.92001)] is improved in this paper. If the parameters of (1) are assumed to be periodic, by using the uniform estimate mentioned above and in terms of principle eigenvalues of certain parabolic equations, the author employs the index technique of himself [SIAM Math. Anal. 32, 504--521 (2000; Zbl 0970.35032)] to give sufficient conditions for the existence of nontrivial periodic solutions of (1).