On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions

Abstract

Convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is shown in situations where the set of stationary solutions creates a C 2 -manifold of finite dimension which is normally stable. We apply the parabolic Hölder setting which allows to deal with nonlocal terms including highest order point evaluation. In this direction some theorems concerning the linearized systems is also extended. As an application of our main result we prove that the lens-shaped networks generated by circular arcs are stable under the surface diffusion flow.

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

“…where B m k is the natural discrete analogue of B k (t m ) and ξ m+ 1 2 k is an appropriate approximation in order to guarantee the unconditional stability for the generalized scheme.…”

“…A model for surface diffusion of a network of curves has been introduced in [41] for d = 2 and generalized to arbitrary space dimensions in [17,32]. Well-posedness was shown in [1] for d = 2 and in [37] for higher space dimensions. We will present the precise mathematical formulation of this evolution law in Section 2 below.…”

Stability analysis for stationary solutions of the Mullins-Sekerka flow with boundary contact

“…We can then express the equations in the fixed reference configuration by means of the Hanzawa transform Θ h , cf. [3], [26], [25], [30], [23], [13], [2]. The transformed system reads as…”

Stability analysis for stationary solutions of the Mullins–Sekerka flow with boundary contact

“…Let us precisely state the problem. We consider a fixed, smooth, and bounded domain in two space dimensions Ω ⊂ ℝ 2 . We assume that the domain can be decomposed as Ω = Ω + (𝑡) ∪ Γ(𝑡) ∪ Ω − (𝑡), where Γ(𝑡) denotes the interior of Γ(𝑡), a smooth one-dimensional curve with two boundary points on 𝜕Ω.…”