On upper semicontinuity of the Allen–Cahn twisted eigenvalues

We give an asymptotic upper bound for the kth twisted eigenvalue of the linearized Allen–Cahn operator in terms of the kth eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103 (2015) 1317–1345).

Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions.

Asymptotic analysis and upper semicontinuity with respect to delay term of attractors to binary mixtures of solids

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Wong–Zakai approximations and limiting dynamics of stochastic Ginzburg–Landau equations

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero.