A conservation law with spatially localized sublinear damping
Besse, Christophe1 ; Carles, Rémi2 ; Ervedoza, Sylvain1
1 Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2 Univ Rennes, CNRS, IRMAR, UMR 6625, F-35000 Rennes, France
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 13-50.
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We consider a general conservation law on the circle, in the presence of a sublinear damping. If the damping acts on the whole circle, then the solution becomes identically zero in finite time, following the same mechanism as the corresponding ordinary differential equation. When the damping acts only locally in space, we show a dichotomy: if the flux function is not zero at the origin, then the transport mechanism causes the extinction of the solution in finite time, as in the first case. On the other hand, if zero is a non-degenerate critical point of the flux function, then the solution becomes extinct in finite time only inside the damping zone, decays algebraically uniformly in space, and we exhibit a boundary layer, shrinking with time, around the damping zone. Numerical illustrations show how similar phenomena may be expected for other equations.

DOI : 10.1016/j.anihpc.2019.03.002
Mots-clés : One-dimensional conservation laws, Sublinear damping, Finite time extinction, Finite time control, Long time behavior
Besse, Christophe 1 ; Carles, Rémi 2 ; Ervedoza, Sylvain 1

1 Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2 Univ Rennes, CNRS, IRMAR, UMR 6625, F-35000 Rennes, France 
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Besse, Christophe; Carles, Rémi; Ervedoza, Sylvain. A conservation law with spatially localized sublinear damping. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 13-50. doi : 10.1016/j.anihpc.2019.03.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.03.002/

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