Long Time Asymptotics For A Class Of Parabolic Equtions

Long time asymptotics studying is the main part in PDE, we can use many techniques such as energy method, Fourier transformation ect. In paper [1], the large time asymptotics for the evolution of a planar curve subjected to the mean curvature flow and constant forcing is studied. It is the study of the motion of a superconducting vortex. When the vortex lies along some line which is a graph in x such that its position may be written as q = (x, v(x, t), 0), the motion satisfies such a equation as following:The aim of this paper is to make the equation above general with the same initial boundary value of the formDepending on the choice of the coefficient functions, we can prove convergence to either a travelling wave or a self-similar profile for large times.