On Convergence of Solutions to Equilibria for Fully Nonlinear Parabolic Systems with Nonlinear Boundary Conditions

Standard Planar Double Bubbles are Stable under Surface Diffusion Flow

“…Appendix E provides for the reader's convenience the derivation in detail. Indeed a similar derivation is done in [1], which is originally given in [10], [5]. Therefore, let us present the final system of fourth-order nonlinear, nonlocal PDEs for t > 0, i = 1, 2, 3 and j = 1, 2, .…”

“…Now we are in a position to look for a suitable PDE theory in order to answer the question of stability. The generalized principle of linearized stability in parabolic Hölder spaces, proved in [1], see also [16,17], provides the tool.…”

“…In this section we present the practical tool, proved in [1], for proving the stability of equilibria of fully nonlinear parabolic systems with nonlinear boundary conditions in situations where the set of stationary solutions creates a C 2manifold of finite dimension which is normally stable. The parabolic Hölder spaces are used as function spaces.…”

“…where A and B is defined in Section 5.1. Due to Remark 2.2 in [1], the spectrum of the linearized operator A 0 consists entirely of eigenvalues. As the analysis of the eigenvalue problem is invariant under the change of variables, we switch to the setting where the functions u i (i = 1, 2, 3) have different domains.…”

“…It is called the generalized principle of linearized stability. This principle is extended in [1] to cover a more general setting. According to this principle, to prove stability, one needs to verify four assumptions known as the conditions of normal stability: (i) the set of stationary solutions creates locally a smooth manifold of finite dimension, (ii) the tangent space of the manifold of stationary solutions is given by the null space of the linearized operator, (iii) the eigenvalue 0 of the linearized operator is semi-simple, (iv) apart from zero, the spectrum of the linearized operator lies in C + .…”