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The paper considers the uniform stabilization of beams with damping acting at an internal point. We study the exponential decay of the solutions in the energy space. Our method is based on multipliers and compactness--uniqueness argument.
We prove regularity results for solutions of nonlinear elliptic equations containing a term which grows as $ |u|^\sigma$, $ \sigma>0$, and satisfies a "sign condition." The source term is supposed to be in $ L^1$.
In this article, we study the network of two neurons with delay. Using the discrete Lyapunov functional of Mallet-Paret and Sell and the techniques developed recently by Krisztin, Walther and Wu (for the scalar case), we obtain a two-dimensional closed disk bordered by a phase-locked periodic orbit and we have a complete description about the structure of various heteroclinic connections in the global forward extension of a three-dimensional $C^1$-submanifold contained in the unstable set of the trivial solution.
We improve on a previous result by Ribaud and Youssfi on existence of self-similar solutions for the nonlinear Schrödinger equation, extending the range of available nonlinearities $\alpha +1 $ to $\alpha$ smaller than 1. We exploit the different behavior of the linear and nonlinear terms.
This paper deals with a singular perturbation problem related to the relaxed exact controllability of a thin shell and its membrane approximation. We point out the subspaces in which we can construct control functions and which allow us to look at the asymptotic limit. Since the problem depends on the geometry of the shell and the selected boundary control action, specific results for elastic hemispherical shells are given.