Existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays
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.
scientific article; zbMATH DE number 1539401
scientific article