Asymptotic Analysis

A Love-Kirchhoff perforated plate or grid with Kelvin-Voigt damping is considered. The grid contains periodic rectangular holes. Detailed mathematical arguments for the derivation of an approximate homogenized model on a domain without perforations are given.

The existence of a maximal global attractor is demonstrated for the dynamical system generated by the initial boundary value problem associated with the motion of a non-linear bipolar fluid; the attractor has the form A=∩_t>0S(t)B^ρ′_(H²(Ω))³ where S(t) is the relevant non-linear semigroup and B^ρ′_(H²(Ω))³ (the ball of radius ρ′>0 in (H²(Ω))³) is, for ρ′ sufficiently large, an absorbing set. Upper bounds are obtained for the Hausdorff and Fractal dimensions of the attractor A.

The Hausdorff dimension of n×m concentration sets is studied. Uniform dimension estimates at each cross section in Rⁿ and Hölder continuity in R^m are used to deduce dimension estimates in R^n+m.

We establish the existence of nontrivial global solutions for the ordinary differential equation u″+f(u)=g(u′), when f is at most linear and g is superlinear. We then provide a result of global uniqueness in certain classes of power-like functions for general autonomous second order ordinary differential equations. This result allows us to show the exceptional character of the global solutions for a class of equations including u″+u=|u′|^p−1u′, for p>1.