The classification of constant weighted curvature curves in the plane with a log-linear density

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $\Bbb R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^y.$