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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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