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

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.

References in corpus (2)

• Inverse problems for two by two reaction-diffusion system using a Carleman estimate with one observation
• A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

Cited by in corpus (1)

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