# On a mixed problem for the parabolic Lam'e type operator

## TL;DR

This paper investigates the ill-posedness of a boundary value problem for the parabolic Lamé operator, a linearized model of compressible fluid flow, and establishes conditions for its uniqueness and solvability using integral representations.

## Contribution

It demonstrates the ill-posedness of the problem in natural and Hölder spaces and provides a uniqueness theorem and solvability conditions via the Integral Representation Method.

## Key findings

- The problem is ill-posed in smooth and Hölder spaces.
- Additional initial data do not ensure well-posedness.
- The Integral Representation Method yields uniqueness and solvability conditions.

## Abstract

We consider a boundary value problem for the parabolic Lam\'e type operator being a linearization of the Navier-Stokes' equations for compressible flow of Newtonian fluids. It consists of recovering a vector-function, satisfying the parabolic Lam\'e type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding H\"older spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the Integral Representation's Method we obtain the Uniqueness Theorem and solvability conditions for the problem.