Radosław Czaja | Searcharxiv

arXiv2026

Solutions of Duhamel's formula with a sum of mappings

Semilinear evolution equations are solved by means of the variation of constants formula under the action of smoothing linear operators and a finite number of nonlinearities operating between chosen Banach spaces of a given family. Admissible spaces of initial conditions are determined for the existence and uniqueness of a suitable notion of solution. Its maximal time of existence, short time and blow up time profiles, global extendibility and regularity are analyzed. For extrapolated fractional power scale of Banach spaces the fulfillment of the corresponding Cauchy problem is shown. Chosen applications to the modified viscous Cahn-Hilliard equation in R^N, the evolution of cell population of varying genotype and the semilinear Schrödinger equation are presented.

math.AP↗

arXiv2025

$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We establish a characterization of both types of manifolds in terms of solutions exhibiting a common growth behavior, analogous to the classical characterization involving hyperbolicity. Furthermore, we introduce a unified formulation of the gap condition, from which known sharp versions are derived. Finally, we show that the constructed inertial manifolds have $C^1$ regularity.

math.DS↗