Gradient flows of time-dependent functionals in metric spaces and applications for PDEs

We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the Wasserstein space $\mathscr{P}_{2}(\mathbb{R}^{d})$ and apply the results for a large class of PDEs with time- dependent coefficients like confinement and interaction potentials and diffusion. Our results can be seen as an extension of those in Ambrosio-Gigli-Savar\'e (2005)[2] to the case of time-dependent functionals. For that matter, we need to consider some residual terms, time-versions of concepts like $\lambda$-convexity, time-differentiability of minimizers for Moreau-Yosida approximations, and a priori estimates with explicit time-dependence for De Giorgi interpolation. Here, functionals can be unbounded from below and satisfy a type of $\lambda$-convexity that changes as the time evolves.