American Institute of Mathematical Sciences

In this paper we study the one-dimensional logarithmic Schrödin-\break ger equation perturbed by an attractive \begin{document}$δ^{\prime}$\end{document}-interaction

\begin{document}$i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log|}}u|2 = 0,(x,t) \in \mathbb{R} \times \mathbb{R} ,$ \end{document}

where $γ>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $δ^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0 < γ≤ 2$, then there is a single ground state and it is an odd function; if $γ>2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.

We investigate the long time behavior of solutions to the differential equation:

\begin{document}$\ddot{x}(t)+\frac{c}{{{\left( 1+t \right)}^{\alpha }}}\dot{x}(t)+\nabla \Phi \left( x(t) \right)=g(t),~t\ge 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ \end{document}

where \begin{document}$c$\end{document} is nonnegative constant, \begin{document} $α∈\lbrack0,1[,Φ \ \ {\rm{is \ \ a}}\ \ C^{1}$\end{document} convex function defined on a Hilbert space \begin{document}$\mathcal{H}$\end{document} and \begin{document} $g∈ L^{1}(0,+∞;\mathcal{H}).$\end{document} We obtain sufficient conditions on the source term \begin{document}$g(t)$\end{document} ensuring the weak or the strong convergence of any trajectory \begin{document} $x(t)$\end{document} of (1) as \begin{document}$t\to ∞$\end{document} to a minimizer of the function \begin{document} $Φ$\end{document} if one exists.

We study an optimal control problem for certain evolution equations in two component composites with \begin{document}$\varepsilon$\end{document}-periodic disconnected inclusions of size \begin{document}$\varepsilon$\end{document} in presence of a jump of the solution on the interface that varies according to a parameter \begin{document}$γ$\end{document}. In particular the case \begin{document}$γ=1$\end{document} is examinated.

In this paper, we discuss the existence of the periodic solutions of a Cahn-Hillard/Allen-Cahn equation which is introduced as a simplification of multiple microscopic mechanisms model in cluster interface evolution. Based on the Schauder type a priori estimates, which here will be obtained by means of a modified Campanato space, we prove the existence of time-periodic solutions in two space dimensions. The uniqueness of solutions is also discussed.

In this paper, we consider a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term acting on the boundary. By using the Faedo-Galerkin approximation method, we first prove the well-posedness of the solutions. By introducing suitable energy and perturbed Lyapunov functionals, we then prove the general decay results, from which the usual exponential and polynomial decay rates are only special cases. To achieve these results, we consider the following two cases according to the coefficient α of the strong damping term: for the presence of the strong damping term (α>0), we use the strong damping term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the strong damping term; for the absence of the strong damping term (α=0), we use the viscoelasticity term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the kernel function.

In this paper we consider a viscoelastic plate equation with a nonlinear weakly dissipative feedback localized on a part of the boundary. Without imposing restrictive assumptions on the boundary frictional damping, we establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.

We consider evolution inclusions driven by a time dependent subdifferential plus a multivalued perturbation. We look for periodic solutions. We prove existence results for the convex problem (convex valued perturbation), for the nonconvex problem (nonconvex valued perturbation) and for extremal trajectories (solutions passing from the extreme points of the multivalued perturbation). We also prove a strong relaxation theorem showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally we present some examples illustrating these results.

A 3D-2D dimension reduction for a nonhomogeneous constrained energy is performed in the realm of $Γ$-convergence, and two-scale convergence for slender domains, providing an integral representation for the limit functional. Applications to supremal functionals are also given.