A conservation law with spatially localized sublinear damping

TL;DR

This paper studies how solutions to a conservation law on a circle with localized sublinear damping either vanish entirely or decay within the damping zone, depending on the flux function's properties, revealing a spatially localized extinction phenomenon.

Contribution

It characterizes the extinction behavior of solutions with localized damping based on the flux function's critical points, extending understanding of damping effects in conservation laws.

Findings

Solutions vanish in finite time if damping is global.

Solutions decay locally with a boundary layer if flux has a non-degenerate critical point.

Numerical simulations support the theoretical phenomena.

Abstract

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.