Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line

Ademir F. Pazoto (1) , Lionel Rosier (2)

1 IF / UFRJ - Instituto de Física da Universidade Federal do Rio de Janeiro
2 IECN - Institut Élie Cartan de Nancy

Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in \cite{linares-pazoto} that the damping is active on a set $(a_0,+\infty)$ with $a_0>0$, we establish the exponential decay of the solutions in the weighted spaces $L^2((x+1)^mdx)$ for $m\in \N ^*$ and $L^2(e^{2bx}dx)$ for $b>0$ by a Lyapunov approach. The decay of the spatial derivatives of the solution is also derived.

Domaines: Equations aux dérivées partielles [math.AP]

Ademir F. Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete and Continuous Dynamical Systems - Series B, 2010, 14 (4), pp.1511-1535. ⟨10.3934/dcdsb.2010.14.1511⟩. ⟨hal-00453183⟩