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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.

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
Full Text: DOI
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