We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved. The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results.
SynopsisWe study the nonlinear equationin R 3 , where A denotes the Laplacian operator, and R and K are real-valued functions satisfying suitable conditions. We use a variational formulation to show the existence of a non-trivial weak solution of the above equation for some real number A. Because of our assumptions on R and K we shall look for solutions which are spherically symmetric, decrease with r = |x| and vanish at infinity.
We consider a dynamical one-dimensional nonlinear von Kármán model depending on one parameter ε > 0 and study its weak limit as ε → 0. We analyse various boundary conditions and prove that the nature of the limit system is very sensitive to them. We prove that, depending on the type of boundary condition we consider, the nonlinearity of Timoshenko's model may vanish.
Abstract.We consider the dynamical von Karman equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially. When the relaxation g satisfies -cig1+p(t) < g\t) <-c0g(t)1+r, 0 < g"(t) < c2g1+i (t), and g,g1+p g i'(K) with p> 2, then the energy decays as n~+t)F • ^ new Liapunov functional is built for this problem.
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