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

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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

  1. Note time-evolution degeneracies can be transformed into singular diffusion, see [49, Problem 3.6].

  2. 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., \(\partial _\rho u:=(|u_x|^{\rho -2}u_x)_x\).

  3. 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.

  4. Note the characteristics (3.27) coincide with the one obtained through our construction, see (3.17). However, the equations for g given by (3.28) and (3.18) are different. This occurs since our Ansatz in (1.11) is different than Matano’s, which is \(L_{pp}=\exp (g)\).

  5. 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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  1. ICMC, Universidade de São Paulo, Av. trab. são-carlense 400, 13566-590, São Carlos, Brazil

    Phillipo Lappicy & Ester Beatriz

  2. Universidad Complutense de Madrid, Pl. de las Ciencias, 3, 28040, Madrid, Spain

    Phillipo Lappicy

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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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