# American Institute of Mathematical Sciences

July  2017, 16(4): 1455-1470. doi: 10.3934/cpaa.2017069

## Damping to prevent the blow-up of the korteweg-de vries equation

 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France

Received  April 2016 Revised  February 2017 Published  April 2017

We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on $\mathscr{L}_{\gamma}$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.

Citation: Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
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##### References:
Initialization
Dichotomy
Initialization
Find the damping
At left, solution at different times $t=$ 0, 2, 4, 4.9925 and 5.3303. At right, $H^1$-norm and $L^2$-norm evolution without damping and a perturbed soliton as initial datum. Here $p=5$
At left, solution at different times $t=$ 0, 2, 5, 10, 11 and 11.3253. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma_k=0.0025$ and a perturbed soliton as initial datum. Here $p=5$
At left, solution at different times $t=$ 0, 2, 5, 10, 15 and 20. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma_k=0.0027$ and a perturbed soliton as initial datum. Here $p=5$
Example of a build damping. Here the initial datum is the perturbed soliton. Here $p=5$
At left, solution at different times $t=$ 0, 2, 5, 10, 15 and 20. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma = \gamma_1$ and a perturbed soliton as initial datum. Here $p=5$
At left, solution at different times $t=$ 0, 2, 5, 7 and 7.928. At right, $H^1$-norm and $L^2$-norm evolution with $\gamma = \gamma_2$ and a perturbed soliton as initial datum. Here $p=5$
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