Spectral analysis for linearizations of the Allen-Cahn equation around rescaled stationary solutions with triple-junction
Tobias Kusche · University of Regensburg Publication Server (University of Regensburg) · 2006
The asymptotic behavior of the vector-valued Allen-Cahn equation was studied by Bronsard and Reitich via formal asymptotic expansion. This leads to vector-valued differential operators, a Sturm-Liouville operator and a Schrödinger operator. The potentials are given by a rescaled statinoary solution with triple-junction. The work investigates the spectrum of the operators in dependence of the asymptotic behavior of the sationary solution. Moreover, we investigate the exponential decay of eigenfunctions which is caused by an operator-valued threshold. In addition to that, we consider realizations of these operators on certain bounded domains, and investigate the smallest eigenvalue in dependence of the domain.