Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation

Abstract

This paper is devoted to studying the 1-D viscous Camassa-Holm equation on a bounded interval. We first deduce the existence and uniqueness of strong solution to the viscous Camassa-Holm equation by using Galerkin method. Then we establish an identity for a second order parabolic operator, by applying this identity we obtain two global Carleman estimates for the linear viscous Camassa-Holm operator. Based on these estimates, we obtain two types of Unique Continuation Property for the viscous Camassa-Holm equation.

Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations

“…with the left-hand side of the first inequality in (12), concluding that (12) also holds for (11). Now, we prove (12) for (13). Indeed, set u = θq, we have…”

“…The main difficulty in establishing global Carleman estimates for the linear stochastic nonclassical diffusion equations is the existence of the term dy xx , this term in the equation introduces significant new technical difficulties, the traditional method does not work in this case. To deal with the term dy xx , we use the method developed in [13].…”

Recurrent solutions of the linearly coupled complex cubic‐quintic Ginzburg‐Landau equations

“…The main idea in this part comes from previous studies. 19,[51][52][53] Step1. Approximate solution.…”