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Coupled nonlinear parabolic systems with time delays. (English) Zbl 0854.35122
There are considered: a bounded domain $$\Omega\subset \mathbb{R}^p$$ $$(p= 1,2,\dots)$$, a finite interval $$[0, T]$$, the sets $$D_T= (0, T]\times \Omega$$, $$S_t= (0, T]\times \partial \Omega$$, $$\overline D_T= [0, T]\times \overline\Omega$$, $$J_i= [-r_i, 0]$$ ($$r_i$$, $$i= 1,\dots, n$$, represent the finite time delays), $$Q^{(i)}_T= [- r_i, T]\times \overline\Omega$$, $$Q_T= Q^{(1)}_T\times\cdots\times Q^{(n)}_T$$, and the system $\partial u_i/\partial t- L_i u_i= f_i(t, x, u(t, x), u_t(t, x)),\quad i= 1,\dots, n,\quad \text{in } D_t,\tag{1}$ where $$u(t, x)\equiv (u_1(t, x),\dots, u_n(t, x))$$, $$u_t(t, x)= (u_1(t- r_1, x),\dots, u_n(t- r_n, x))$$ ($$u_i(t, x)$$ is a density function), and for each $$i$$, $$L_i$$ is a uniformly elliptic operator given in the form $L_i u_i\equiv \sum^p_{j, k= 1} a^{(i)}_{j,k}(t, x) \partial^2 u_i/\partial x_j \partial x_k + \sum^p_{j= 1} b^{(i)}_j (t, x) \partial u_i/\partial x_j.$ Some boundary and initial conditions are also given on $$S_T$$ and $$J_i\times \Omega$$, $$i= 1,\dots, n$$. The functions $$f_i(t, x, u, v)$$ are Hölder continuous in $$(t, x)$$ and locally Lipschitz continuous in $$(u, v)$$. In addition to the system (1) the system $\partial u_i/\partial t- L_i u_i= f_i(t, x, u(t, x), u_t(t, x))+ \int_\Omega g_i(t, x, x', u(t, x'), u_t(t, x')) dx',\quad i= 1,\dots, n,\tag{2}$ is also considered, where $$g_i$$ satisfy the same properties as $$f_i$$. A monotone iterative scheme, using upper-lower solutions, is studied and some existence-comparison theorems for (1) and (2) are given. Some applications to several model problems arising from ecology and nuclear engineering are presented.

##### MSC:
 35R10 Partial functional-differential equations 35K57 Reaction-diffusion equations 45K05 Integro-partial differential equations
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