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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.
##### MSC:
 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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