A Lyapunov function for fully nonlinear parabolic equations in one spatial variable

Lappicy, Phillipo; Fiedler, Bernold

doi:10.1007/s40863-018-00115-2

Cited by 14 publications

(12 citation statements)

References 17 publications

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“…We firstly introduce the functional setting in Section 2.1, and construct a Lyapunov functional for the singular case by modifying Matano's arguments from [27], and its generalization for fully nonlinear equations [24]. In particular this implies that the attractor consists of equilibria and heteroclinics.…”

Section: Resultsmentioning

confidence: 99%

Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

Lappicy

2018

J Dyn Diff Equat

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The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovský and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equations by the author (2017). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is computed yielding the well known Chafee-Infante attractor.Keywords: parabolic equations, singular coefficients, infinite dimensional dynamical systems, global attractor, Sturm attractor. 0 < ǫ ≤ a(x, u, p) ≤ δ 1. the global attractor A of (1.2) consists of finitely many equilibria E and their heteroclinic orbits H.2. there exists a heteroclinic u(t) ∈ H between u − , u + ∈ E such thatif, and only if, u − and u + are adjacent and i(u − ) > i(u + ).The first claim follows due to the existence of a Lyapunov functional constructed by Matano [29] and Zelenyak [44]. A modification of such functional for the case of singular coefficients is done in Section 2.1.

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Section: Resultsmentioning

confidence: 99%

Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

Lappicy

2018

J Dyn Diff Equat

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“…To construct a Lyapunov function E as in (1.4), we rewrite the fully nonlinear parabolic equation (1.1) suitably, following the spirit of [19]. Then we continue with a modification of Matano's original idea for non-degenerate quasilinear equations in order to incorporate degeneracies.…”

Section: Resultsmentioning

confidence: 99%

“…See also [10] for concise expositions of Matano's method. This method was extended to fully nonlinear non-degenerate parabolic equations, when f q • f r < 0, in [19]. An analogous method for Jacobi systems, a spatially discrete variant, was developed in [11].…”

Section: Resultsmentioning

confidence: 99%

An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension

Lappicy¹,

Beatriz²

2022

Preprint

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Energy (or Lyapunov) functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations with gradient dependency. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. In particular, we provide a new energy formula for the porous medium equation.

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“…It is known that (1.1) possesses a Lyapunov function, alias a variational or gradientlike structure, under separated boundary conditions; see [Ze68,Ma78,MaNa97,Hu11,Fietal14,LaFi18]. In particular, the global attractor consists of equilibria and of solutions u(t, ·), t ∈ R, with forward and backward limits, i.e.…”

Section: Introductionmentioning

confidence: 99%