Exponential stabilization of a coupled system of Korteweg-de Vries equations with localized damping.
The authors study following initial-boundary value problem for a coupled system of Korteweg-de Vries equations \[ \begin{aligned} u_t+u_x+u_{xxx}+a_3v_{xxx}+uu_x+a_1vv_x+a_2(uv)_x+\lambda(x)u&=0,\\ b_1v_t+v_x+v_{xxx}+b_2a_3u_{xxx}+vv_x+b_2a_2uu_x+b_2a_1(uv)_x+\lambda(x)v&=0, \end{aligned} \] where \(0<x<L\) and \(t>0\); \[ \begin{aligned} u(0,t)=u(L,t)=0, \quad v(0,t)=v(L,t)=0 &\quad \text{for all }t>0, \\ u_x(L,t)=0, \quad v_x(L,t)=0 &\quad \text{for all } t>0,\\ u(x,0)=u_0(x), \quad v(x,0)=v_0(x), &\quad\text{for } 0<x<L.\end{aligned} \] Here \(a_1,a_2,a_3,b_1\) and \(b_2\) are real constants with \(b_1>0\), \(b_2>0\) and \(\lambda=\lambda(x)\) is a nonnegative given function belonging to \(L^\infty(0,L)\), \(\lambda(x)\geq\lambda_0>0\) almost everywhere in an open non-empty proper subset of \((0,L)\). The main result of this paper says that the total energy \(E(t)=\frac{1}{2}\int_0^L (u^2 + b_1 v^2)\,dx\) associated with the problem decays exponentially as \(t\rightarrow +\infty\) for bounded sets of initial data. The result is proved assuming that \(b_2=1\) and \(0<a_3<1\).
initial-boundary value problem for a coupled system of Korteweg-de Vries equations
scientific article; zbMATH DE number 2002537