Published Paper

Inserted: 18 jan 2024
Last Updated: 10 jun 2025

Journal: J. Funct. Anal.
Volume: 289
Number: 8
Pages: 111067
Year: 2025
Doi: https://doi.org/10.1016/j.jfa.2025.111067

Abstract:

We discuss a variational approach to doubly nonlinear wave equations of the form $\rho u_{tt} + g (u_t) - \Delta u + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit $\rho \to 0$, both at the functional level and on that of the equation.