Traveling wave solutions of the heat flow of director fields. (Fronts progressifs pour le flot des champs de vecteurs unitaires.) (English) Zbl 1176.35093
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.
\[ h_{rr}+h_{zz}+\frac{h_r}{r}+ch_z-\frac{\sin(2h)}{2r^2}=0, \quad r\in{(0,1)},\;z\in{\mathbb{R}};\qquad h(1,z)=g(z), \]
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.
Reviewer: Wuqing Ning (Hefei)
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35A21 | Singularity in context of PDEs |
| 35A15 | Variational methods applied to PDEs |
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