An inverse problem for an abstract evolution equation

TL;DR

This paper investigates an inverse problem for an abstract evolution equation, focusing on determining a coefficient from additional data, and reduces it to a Volterra integral equation with applications to parabolic PDEs.

Contribution

It introduces a method to solve the inverse coefficient problem by reducing it to a Volterra equation, with specific applications to parabolic differential equations.

Findings

Reduction of the inverse problem to a Volterra equation of the second kind.

Application of the method to parabolic equations with second order operators.

Provides a framework for coefficient identification in evolution equations.

Abstract

The inverse problem of finding the coefficient $\g$ in the equation $\dot{u}=A(t)u+\g(t)u+f(t)$ from the extra data of the form $\phi(t)=u(t),w$ is studied. The problem is reduced to a Volterra equation of the second kind. Applications are given to parabolic equations with second order differential operators.