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We consider the parabolic equation
(P) , (t,x) ∈ ℝ₊ × ℝⁿ,
and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained extend...
The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical...
Consider the ordinary differential equation
(1) ẋ = Lx + K(x)
on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends...
The author examined non-zero -periodic (in time) solutions for a semilinear beam equation under the condition that the period is an irrational multiple of the length. It is shown that for a.e. (in the sense of the Lebesgue measure on ) the solutions do exist provided the right-hand side of the equation is sublinear.
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