# Asymptotic behavior of nonautonomous monotone and subgradient evolution equations

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Abstract : In a Hilbert setting $H$, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations $\dot x(t)+A_t(x(t))\ni 0$, where for each $t\geq 0$, $A_t:H\rightrightarrows H$ denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory $x(\cdot )$ to a zero of a limit maximal monotone operator $A_\infty$ as the time variable $t$ tends to $+\infty$. The crucial point is to use the Brézis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of $\mathrm {gph} A_\infty$ over $\mathrm {gph} A_t$ tends to zero. This approach gives a sharp and unifying view of this subject. In the case of operators $A_t= \partial \phi _t$ which are subdifferentials of proper closed convex functions $\phi _t$, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions. Illustrations are given in the field of coupled systems and partial differential equations.
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Journal articles

https://hal.archives-ouvertes.fr/hal-01660954
Contributor : Imb - Université de Bourgogne <>
Submitted on : Monday, December 11, 2017 - 3:29:14 PM
Last modification on : Tuesday, May 28, 2019 - 1:54:03 PM

### Citation

Hedy Attouch, Alexandre Cabot, Marc-Olivier Czarnecki. Asymptotic behavior of nonautonomous monotone and subgradient evolution equations. Transactions of the American Mathematical Society, American Mathematical Society, 2018, 370 (2), pp.755-790. ⟨10.1090/tran/6965 ⟩. ⟨hal-01660954⟩

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