Abstract
Energy (or Lyapunov) functions are used 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. We modify Matano’s method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation.
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Notes
Note time-evolution degeneracies can be transformed into singular diffusion, see [49, Problem 3.6].
In the literature, this operator is called the p-Laplacian. However, in our notation \(p:=u_x\) and thus we replace the parameter p by \(\rho \) in the degenerate diffusion operator, i.e., \(\partial _\rho u:=(|u_x|^{\rho -2}u_x)_x\).
For \(p_0= 0\) (i.e. \(p\equiv 0\) and \(g\equiv g_0\)), we obtain that \(L_{pp} = \exp (g_0)/(1 + p^2)^{\frac{3}{2}}\) for all \(n\in {\mathbb {N}}\). This yields \(E = \int _0^1 \sqrt{1+u_x^2} \, dx\), up to a multiplicative constant \(\exp (g_0)\), which is the perimeter of the curve; similar to the mean curvature flow with Hamiltonian forcing in Sect. 3.1.
The diffusion given by \((|u|^{m-1}u)_{xx}=m|u|^{m-1}u_{xx}+m(m-1)|u|^{m-3}u u_x^2\) is a natural extension that takes sign-changing solutions into account, which is thereby called the signed PME in [49]. For the sake of simplicity, we proceed with the non-signed PME in the main text.
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Acknowledgements
We are grateful for the suggestions of Jia-Yuan Dai that significantly improved our paper. PL was firstly supported by FAPESP, 17/07882-0, and later on by Marie Skłodowska-Curie Actions, UNA4CAREER H2020 Cofund, 847635, with the project DYNCOSMOS. EB was firstly supported by CNPq, 135896/2020-7, and later on by FAPESP, 23/07941-7.
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Lappicy, P., Beatriz, E. An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension. Math. Ann. 389, 4125–4147 (2024). https://doi.org/10.1007/s00208-023-02740-5
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DOI: https://doi.org/10.1007/s00208-023-02740-5