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Invariant sets and connecting orbits for nonlinear evolution equations at resonance. (English) Zbl 1363.37005
The semilinear equations \[ \begin{aligned} &u_t(t,x)=\Delta u(t,x)+\lambda u(t,x)+ f(x,u(t,x)),\\ &u_{tt}(t,x)=\Delta u(t,x)+ c\Delta u_t(t,x)+\lambda u(t,x)+f(x,u(t,x))\end{aligned} \] are considered. Here \(c>0\), \(\lambda\in\mathbb{R}\), \(\Delta\) is the Laplacian with Dirichlet boundary conditions, \(f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}\) is continuous and bounded, and \(\Omega\subset\mathbb{R}^n\) is an open and bounded set with the boundary of \(C^\infty\)-class. Under several conditions imposed on the equations, including some geometrical assumptions which generalize the Landesman-Lazer conditions and the conditions of strong resonance at infinity, the main theorems of the paper assert the existence of non-zero compact orbits accumulating at \(0\) as time tends to \(\infty\) or to \(-\infty\). The proofs of the theorems are based on previous publications of the author, in which the Conley index formulas were determined under the geometrical assumptions.
37B30 Index theory for dynamical systems, Morse-Conley indices
35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35L71 Second-order semilinear hyperbolic equations
35P05 General topics in linear spectral theory for PDEs
35L10 Second-order hyperbolic equations
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