Recovering Initial States in Certain Quasilinear Parabolic Problems from Time Averages

TL;DR

This paper demonstrates that, under specific conditions, the initial state of certain quasilinear parabolic equations can be uniquely reconstructed from small time averages, with applications to chemotaxis and reaction-diffusion models.

Contribution

It establishes a novel uniqueness result for recovering initial states from time averages in quasilinear parabolic problems under new regularity and growth conditions.

Findings

Unique recovery of initial states from small time averages.

Applicability to chemotaxis models.

Applicability to reaction-diffusion systems.

Abstract

The inverse problem of reconstructing the initial state in quasilinear parabolic equations from time averages is investigated. Under suitable regularity assumptions on the quasilinear structure and a superlinear growth condition near zero for the semilinear part, it is shown that the initial state can be uniquely recovered from small time averages taken over an arbitrary time period. The applicability of the result is demonstrated for certain chemotaxis models and reaction-diffusion systems.