Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement

TL;DR

This paper addresses the inverse problem of reconstructing an unknown convolution kernel in a semilinear parabolic PDE using a global measurement, establishing theoretical results and proposing a convergent numerical algorithm.

Contribution

It introduces a novel numerical method based on Rothe's approach for recovering the kernel and proves its convergence, along with existence, uniqueness, and regularity results.

Findings

Proved existence, uniqueness, and regularity of solutions.

Developed a convergent numerical algorithm for kernel reconstruction.

Validated the method through theoretical analysis.

Abstract

A semilinear parabolic problem of second order with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe's method, derive a priori estimates and prove convergence of iterates towards the exact solution.