Nonlinear stability analysis of a two-dimensional diffusive free boundary problem (Q605502)

Summary: We explore global existence and stability of planar solutions to a multi-dimensional Case II polymer diffusion model which takes the form of a one-phase free boundary problem with phase onset. Due to a particular boundary condition, convergence cannot be expected on the whole domain. A boundary integral formulation derived in [\textit{M. Webster} and \textit{P. Guidotti}, Discrete Contin. Dyn. Syst. 26, No.~2, 713--736 (2010; Zbl 1186.35249)] is shown to remain valid in the present context and allows us to circumvent this difficulty by restricting the analysis to the free boundary. The integral operators arising in the boundary integral formulation are analyzed by methods of pseudodifferential calculus. This is possible as explicit symbols are available for the relevant kernels. Spectral analysis of the linearization can then be combined with a known principle of linearized stability [\textit{A. Lunardi}, Analytic semigroups and optimal regularity in parabolic problems, Basel: Birkhäuser (1995; Zbl 0816.35001)] to obtain local exponential stability of planar solutions with respect to two-dimensional perturbations.