On a perturbed functional integral equation of Urysohn type

We study the existence of monotonic solutions for a perturbed functional integral equation of Urysohn type in the space of Lebesgue integrable functions on an unbounded interval. The technique associated with measures of noncompactness (in both the weak and the strong sense) and the Darbo fixed point are the main tool to prove our main result

Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well

Transmission conditions obtained by homogenisation

Given a bounded open set in Rn, n 652, and a sequence (Kj) of compact sets converging to an (n-1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on \u3a9\Kj, with Neumann boundary conditions on 02(\u3a9\Kj). We prove that the limit of these solutions is a minimiser of the same functional on \u3a9\M subjected to a transmission condition on M, which can be expressed through a measure \ub5 supported on M. The class of all measures that can be obtained in this way is characterised, and the link between the measure \ub5 and the sequence (Kj) is expressed by means of suitable local minimum problems