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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.
Besse, Christophe 1 ; Carles, Rémi 2 ; Ervedoza, Sylvain 1
@article{AIHPC_2020__37_1_13_0,
author = {Besse, Christophe and Carles, R\'emi and Ervedoza, Sylvain},
title = {A conservation law with spatially localized sublinear damping},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {13--50},
publisher = {Elsevier},
volume = {37},
number = {1},
year = {2020},
doi = {10.1016/j.anihpc.2019.03.002},
mrnumber = {4049915},
zbl = {1430.35154},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.03.002/}
}
TY - JOUR AU - Besse, Christophe AU - Carles, Rémi AU - Ervedoza, Sylvain TI - A conservation law with spatially localized sublinear damping JO - Annales de l'I.H.P. Analyse non linéaire PY - 2020 SP - 13 EP - 50 VL - 37 IS - 1 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.03.002/ DO - 10.1016/j.anihpc.2019.03.002 LA - en ID - AIHPC_2020__37_1_13_0 ER -
%0 Journal Article %A Besse, Christophe %A Carles, Rémi %A Ervedoza, Sylvain %T A conservation law with spatially localized sublinear damping %J Annales de l'I.H.P. Analyse non linéaire %D 2020 %P 13-50 %V 37 %N 1 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.03.002/ %R 10.1016/j.anihpc.2019.03.002 %G en %F AIHPC_2020__37_1_13_0
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
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