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Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays. (English) Zbl 0970.35004
There are studied nonlinear systems of parabolic equations with delays of the form: $\partial u_i/\partial t- L_iu_i= f_i(x, t,u,u_\sigma)\quad\text{on } \Omega\times (0,\infty),$
$B_i[u_i](x,t)= h_i(x,t)\quad\text{on }\partial\Omega\times (0,\infty),$ $$u_i(x,t)= \mu_i(x,t)$$ on $$\Omega\times [-\sigma_i, 0]$$, where $$u(x,t)= (u_1(x, t),\dots, u_n(x, t))$$, $$u_\sigma(x,t)= (u_1(x, t-\sigma_1),\dots, u_n(x, t-\sigma_n))$$; $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ with the boundary $$\partial\Omega$$; $$\sigma_1,\dots,\sigma_n$$ are positive constants $L_iu_i\equiv \sum^N_{j,k= 1} a^i_{jk}(x, t)\partial^2 u_i/\partial x_j\partial x_k+ \sum^N_{j=1} b^i_j(x, t)\partial u_i/\partial x_j,$ (for each $$t\in \mathbb{R}$$, $$L_i$$ is a uniformly elliptic operator); $$B_iu_i\equiv \alpha_i\partial u_i/\partial n+ \beta_i(x, t)u_i$$ ($$\alpha_i=0$$, $$\beta_i(x,t)= 1$$ or $$\alpha_i(x,t)= 1$$, $$\beta_i(x, t)\geq 0$$ and it is allowed to be of different type for different $$i$$); the functions $$a^i_{jk}$$, $$h_i$$, $$\beta_i$$ belong, respectively, to some spaces of $$T$$-periodic, in $$t$$, functions; $$f_i$$, are $$T$$-periodic in $$t$$; $$\eta_i$$ are Hölder continuous.
The authors obtain the existence of periodic quasisolutions and some asymptotic properties; some applications to some models in ecology are given to illustrate the obtained results.

##### MSC:
 35B10 Periodic solutions to PDEs 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35R10 Partial functional-differential equations 35K55 Nonlinear parabolic equations 92D40 Ecology
##### Keywords:
asymptotic properties
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