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

Pierre Garnier

doi:10.3934/cpaa.2017069

Article Contents

2017, Volume 16, Issue 4: 1455-1470. Doi: 10.3934/cpaa.2017069

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

Pierre Garnier

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 online: July 2017

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Abstract

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.

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Mathematics Subject Classification: 35B44, 35Q53, 76B03, 76B15.

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Figure 1. Initialization

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Figure 2. Dichotomy

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Figure 3. Initialization

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Figure 4. Find the damping

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Figure 5. 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 $p=5$

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Figure 6. 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 $p=5$

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Figure 7. 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 $p=5$

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Figure 8. Example of a build damping. Here the initial datum is the perturbed soliton. Here $p=5$

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Figure 9. 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 $p=5$

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Figure 10. 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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References

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