A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain $\Omega\times {\mathcal O}$ of $\R^{M+N}$

TL;DR

This paper investigates a highly ill-posed degenerate parabolic problem in an unbounded domain with unbounded coefficients, establishing conditions for uniqueness and continuous dependence on data, and applying results to a related integrodifferential equation.

Contribution

It provides new sufficient conditions for uniqueness and stability of solutions to a strongly ill-posed degenerate parabolic problem with unbounded coefficients.

Findings

Established conditions ensuring uniqueness of solutions.

Identified metrics for continuous dependence on data.

Extended results to a convolution integrodifferential equation.

Abstract

In this paper we deal with a strongly ill-posed second-order degenerate parabolic problem in the unbounded open set $\Omega\times {\mathcal O}\subset \mathbb R^{M+N}$, related to a linear equation with unbounded coefficients, with no initial condition, but endowed with the usual Dirichlet condition on $(0,T)\times \partial(\Omega\times {\mathcal O})$ and an additional condition involving the $x$-normal derivative on $\Gamma\times {\mathcal O}$, $\Gamma$ being an open subset of $\Omega$. The task of this paper is twofold: determining sufficient conditions on our data implying the uniqueness of the solution $u$ to the boundary value problem as well as determining a pair of metrics with respect of which $u$ depends continuously on the data. The results obtained for the parabolic problem are then applied to a similar problem for a convolution integrodifferential linear parabolic equation.