Steady state solution to the weakly damped forced Korteweg-de Vries equation (Q1288380)

The authors consider the nonlinear third-order periodic boundary value problem \[ u'''(x)+ \gamma u(x)+ \beta u(x)u'(x)= f(x),\quad u(0)-u(1)= u'(0)-u'(1)= u''(0)-u''(1)= 0,\tag{1} \] where \(\gamma>0\) and \(\beta\) are real parameters. Note that (1) represents steady state solutions the weakly damped forced KdV equation \(u_t+u_{xxx} +\gamma u+\beta uu_x=f(x)\) \(x\in I:=[0,1]\). For \(f\in L^2(I)\) the problem (1) has at least one solution. Moreover, if \(|\beta| <2\pi^2/M_0\) for some constant \(M_0\), then the solution is unique. The proof relies on the topological transversality theorem of Granas. It is shown that every solution of the weakly damped forced KdV equation converges to the steady state solution as \(t\to\infty\).