Optimal systems and invariant solutions for the curve shortening problem (Q1866511)

The authors give a systematic investigation of the group invariant solutions of the generalized curve shortening problem described by the following non-uniformly parabolic equation \[ u_t=\left|\frac{u_{xx}}{(1+u^{2}_{x})^{3/2}}\right|^{\sigma-1} \frac{u_{xx}}{(1+u^{2}_{x})}, \quad \sigma>0 \] For this equation they determine the Lie algebra of all infinitesimal symmetries and show that they form a basis of all symmetries if \(\sigma\not =1/3\). When \(\sigma=1/3\) the Lie algebra of infinitesimal symmetries is of dimension seven, two more than for the non-affine case \(\sigma=1/3\). To describe all group invariant solutions an optimal system of subalgebras is determined.