Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

TL;DR

This paper establishes conditions for invariance in nonlinear evolution problems governed by m-accretive operators, with applications to obstacle problems and population models, extending previous results and developing new criteria for strict invariance.

Contribution

It provides new sufficient conditions for invariance and strict invariance in nonlinear evolution problems, including non-reflexive Banach spaces, improving prior work.

Findings

Derived invariance conditions using Dini derivatives.

Extended approach to non-reflexive Banach spaces.

Applied results to obstacle problems and population models.

Abstract

Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) $m$-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint functional are expressed in terms of the Dini derivative of that functional, outside the considered sublevel set in directions determined by the governing $m$-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper [P. Cannarsa, G. Da Prato, H. Frankowska, Invariance of quasi-dissipative systems in Banach spaces. J. Math. Anal. App. 457 (2018), 1173-1187] is presented. Applications to nonlinear obstacle problems and age-structured population models are presented in spaces of continuous functions where advantages of that approach are taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct applications to problems with barriers are shown.