Recovering Initial States in Semilinear Parabolic Problems from Time-Averages

TL;DR

This paper demonstrates how to recover initial states in semilinear parabolic problems with superlinear nonlinearities using small time-averages, leveraging well-posedness in time-weighted spaces.

Contribution

It introduces a method to recover initial states from time-averages in semilinear parabolic problems with superlinear nonlinearities, based on well-posedness results.

Findings

Successful recovery of initial states from small time-averages

Application of well-posedness in time-weighted spaces

Extension to nonlinearities of superlinear behavior

Abstract

Well-posedness of certain semilinear parabolic problems with nonlocal initial conditions is shown in time-weighted spaces. The result is applied to recover the initial states in semilinear parabolic problems with nonlinearities of superlinear behavior near zero from small time-averages over arbitrary time periods.