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The trajectory attractor of a nonlinear elliptic system in a cylindrical domain. (English. Russian original) Zbl 0871.35016

Sb. Math. 187, No. 12, 1755-1789 (1996); translation from Mat. Sb. 187, No. 12, 21-56 (1996).
Summary: In the half-cylinder \(\Omega_+ =\mathbb{R}_+ \times\omega\), \(\omega\subset\mathbb{R}^n\), we study a second-order system of elliptic equations containing a nonlinear function \(f(u,x_0,x') =(f^1, \dots, f^k)\) and right-hand side \(g(x_0,x') =(g^1, \dots, g^k)\), \(x_0\in \mathbb{R}_+\), \(x'\in \omega\). If these functions satisfy certain conditions, then it is proved that the first boundary-value problem for this system has at least one solution belonging to the space \([H_{2,p}^{\text{loc}} (\Omega_+)]^k\), \(p>n+1\). We study the behaviour of the solutions \(u(x_0,x')\) of this system as \(x_0\to +\infty\). Along with the original system we study the family of systems obtained from it through shifting with respect to \(x_0\) by all \(h\), \(h\geq 0\). A semigroup \(\{T(h),\;h\geq 0\}\), \(T(h)u(x_0, \cdot)= u(x_0+h, \cdot)\) acts on the set of solutions \(K^+\) of these systems of equations. It is proved that this semigroup has a trajectory attractor \(\mathbb{A}\) consisting of the solutions \(v(x_0,x')\) in \(K^+\) that admit a bounded extension to the entire cylinder \(\Omega= \mathbb{R}\times \omega\). Solutions \(u(x_0,x') \in K^+\) are attracted by the attractor \(\mathbb{A}\) as \(x_0\to +\infty\). We give a number of applications and consider some questions of the perturbations of the original system of equations.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
35J45 Systems of elliptic equations, general (MSC2000)
35B35 Stability in context of PDEs
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