Traveling wave solutions of the heat flow of director fields
The authors consider the heat flow of harmonic maps from infinite cylinder \(\{(x_1,x_2,x_3)\in\mathbb R^3: x_1^2+x_2^2<1\}\) to the unit sphere \(\mathbb S^2: u_t=\Delta u+|\nabla u|^2u\), to which by cylindrical coordinates there corresponds a scalar angle function \(h(r,x_3,t)\). In order to study nonuniqueness of axially symmetric solutions of the heat flow, they look for traveling wave solutions \(h(r,x_3,t)=h(r,z), z=x_3-ct\) which satisfy the following singular elliptic equation \[ h_{rr}+h_{zz}+\frac{h_r}{r}+ch_z-\frac{\sin(2h)}{2r^2}=0, \quad r\in{(0,1)},\;z\in{\mathbb{R}};\qquad h(1,z)=g(z), \] where \(c>0\) is the wave speed and \(g\) is a given function satisfying some special conditions. By the variation methods and the concept of relaxed energy, they prove in two cases the existence of traveling wave solutions with point singularity of topological degree 0 or 1.
scientific article; zbMATH DE number 5519838
scientific article