In this article we discuss some estimates of the number of the negative eigenvalues and their moments of energy for an elliptic operator L = L0 - V(x) defined in Hm(R+n) with the Robin boundary conditions containing a potential W(x), in terms of some integrals of V and W.

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µin three dimensions, $\mathrm{div}\left(\mu (\nabla u+\nabla u^t)\right)+\nabla \left(\lambda \mathrm{div}u\right)+Vu=0$ whereλ and μ are Lipschitz continuous and V∈L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µ in three dimensions, $\mathrm{div}\left(\mu (\nabla u+\nabla u^t)\right)+\nabla \left(\lambda \mathrm{div}u\right)+Vu=0$ where λ and μ are Lipschitz continuous and V∈L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.

We prove the following unique continuation property. Let $u$ be a solution of a second order linear parabolic equation and $S$ a segment parallel to the $t$-axis. If $u$ has a zero of order faster than any non constant and time independent polynomial at each point of $S$ then $u$ vanishes in each point, $\left(x,t^{\prime }\right)$, such that the plane $t=t^{\prime }$ has a non empty intersection with $S$.

This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining...

This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining...

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition$$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|\<r}|x-y{|}^{2-n}v\left(y\right)dy=0$$for all compact $K\subset \Omega$, where if $n=2$, we replace $|x-y{|}^{2-n}$ by $-log|x-y|$. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$$$...$$

We prove unique continuation for solutions of the inequality $\left|\Delta u\right(x\left)\right|\le V\left(x\right)\left|\nabla u\right(x\left)\right|$, $x\in \Omega$ a connected set contained in ${\mathbf{R}}^n$ and $V$ is in the Morrey spaces $F^{\alpha ,p}$, with $p\ge (n-2)/2(1-\alpha )$ and $\alpha \<1$. These spaces include $L^q$ for $q\ge (3n-2)/2$ (see [H], [BKRS]). If $p=(n-2)/2(1-\alpha )$, the extra assumption of $V$ being small enough is needed.

Much of this paper will be concerned with the proof of the followingTheorem 1. Suppose d ≥ 3, r = max {d, (3d - 4)/2}. If V ∈ Llocr(Rd), then the differential inequality |Δu| ≤ V |∇u| has the strong unique continuation property in the following sense: If u belongs to the Sobolev space Wloc2,p and if |Δu| ≤ V |∇u| andlimR→0 R-N ∫|x| < R |∇u|p' = 0for all N then u is constant.