Special Session 47: Singular limits in elliptic and parabolic PDEs

On the stability of radial solutions to an anisotropic Ginzburg-Landau equation

Andres Zuniga
University of O`Higgins (UOH)
Co-Author(s):    Xavier Lamy (Universite Paul Sabatier, Toulouse, France)
We study the linear stability of entire radial solutions $u(re^{i\theta})=f(r)e^{i\theta}$, with positive increasing profile $f(r)$, to the anisotropic Ginzburg-Landau equation $-\Delta u-\delta(\partial_x+i\partial_y)^2\bar{u}=(1-|u|^2)u$, which arises in various liquid crystal models. In the isotropic case $\delta=0$, Mironescu showed that such solution is nondegenerately stable. We prove stability of this radial solution in the range $\delta\in(\delta_1,0]$ and instability outside this range. In strong contrast with the isotropic case, stability with respect to higher Fourier modes is not a direct consequence of stability with respect to lower Fourier modes. In particular, in the case where the anisotropy parameter is close to $-1$, lower modes are stable and yet higher modes are unstable.