On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions. (English) Zbl 1366.35006
The authors’ motivation for this work is the appearance of nonlocal, nonlinear terms together with general nonlinear boundary conditions when studying stability for the surface diffusion flow with triple junctions. The general setting is a nonlinear parabolic system of order \(2m\) coupled with general nonlinear boundary conditions in parabolic spaces.
Suppose such a boundary value problem has a finite dimensional \(C^2\) manifold of equilibria \(\mathcal{E}\) such that at a point \(u_*\in \mathcal{E}\) the null space \(N(A_0)\) of the linearization \(A_0\) is given by the tangent space of \(\mathcal{E}\) at \(u_*\), zero is a semi-simple eigenvalue of \(A_0\), and the rest of the spectrum of \(A_0\) is stable. Their main result states that under these assumptions, solutions with initial data close to \(u_*\) exist globally in the classical sense and converge towards the manifold of equilibria as \(t\rightarrow \infty\).
This generalized principle of linearized stability was introduced by J. Prüss et al. [J. Differ. Equations 246, No. 10, 3902–3931 (2009; Zbl 1172.35010)] for abstract quasilinear problems, and also for vector-valued quasilinear parabolic systems with nonlinear boundary conditions in the framework of \(L_p\)-optimal regularity. It was then extended by them to cover a wider range of settings [J. Prüss et al., Discrete Contin. Dyn. Syst. 2009, 612–621 (2009; Zbl 1194.35047)], including fully nonlinear abstract evolution equations without boundary conditions.
The authors applied their result in [“Standard planar double bubbles are stable under surface diffusion flow”, Preprint, arXiv:1505.02979] to show that stationary solutions of the form of the standard planar double bubble are stable under the surface diffusion flow.
As a further application, they show here that lens-shaped networks generated by circular arcs are stable under surface diffusion flow. These are the simplest examples of more general triple junctions where the resulting partial differential equations have nonlocal terms in the highest order derivatives.
Suppose such a boundary value problem has a finite dimensional \(C^2\) manifold of equilibria \(\mathcal{E}\) such that at a point \(u_*\in \mathcal{E}\) the null space \(N(A_0)\) of the linearization \(A_0\) is given by the tangent space of \(\mathcal{E}\) at \(u_*\), zero is a semi-simple eigenvalue of \(A_0\), and the rest of the spectrum of \(A_0\) is stable. Their main result states that under these assumptions, solutions with initial data close to \(u_*\) exist globally in the classical sense and converge towards the manifold of equilibria as \(t\rightarrow \infty\).
This generalized principle of linearized stability was introduced by J. Prüss et al. [J. Differ. Equations 246, No. 10, 3902–3931 (2009; Zbl 1172.35010)] for abstract quasilinear problems, and also for vector-valued quasilinear parabolic systems with nonlinear boundary conditions in the framework of \(L_p\)-optimal regularity. It was then extended by them to cover a wider range of settings [J. Prüss et al., Discrete Contin. Dyn. Syst. 2009, 612–621 (2009; Zbl 1194.35047)], including fully nonlinear abstract evolution equations without boundary conditions.
The authors applied their result in [“Standard planar double bubbles are stable under surface diffusion flow”, Preprint, arXiv:1505.02979] to show that stationary solutions of the form of the standard planar double bubble are stable under the surface diffusion flow.
As a further application, they show here that lens-shaped networks generated by circular arcs are stable under surface diffusion flow. These are the simplest examples of more general triple junctions where the resulting partial differential equations have nonlocal terms in the highest order derivatives.
Reviewer: John Urbas (Canberra)
MSC:
| 35B35 | Stability in context of PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35K52 | Initial-boundary value problems for higher-order parabolic systems |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
fully nonlinear parabolic systems; general nonlinear boundary conditions; normally stable; surface diffusion flow; triple junctions; lens-shaped networkReferences:
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