Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component

Abstract

This article presents an inverse source problem for a cascade system of n coupled degenerate parabolic equations. In particular, we prove stability and uniqueness results for the inverse problem of determining the source terms by observations in an arbitrary subdomain over a time interval of only one component and data of the n components at a fixed positive time T over the whole spatial domain. The proof is based on the application of a Carleman estimate with a single observation acting on a subdomain.

Null controllability for degenerate parabolic equations with a nonlocal space term

“…Remark 6. Note that, in [3] Theorem 3.6 is shown for θ(t) = 1 t 4 (T −t) 4 . However, by [1, Remark 1], one can prove that the result remains true also for θ…”

“…Indeed, for instance, in the absence of the nonlocal term, the equation y t − ay xx = 0 can be rewritten as y t − (ay x ) x + a x y x = 0 (3) only if a x exists. Moreover, as described in [8], degenerate equations of the form (3), are well-posed in L 2 (0, 1) under the structural assumption…”

Controllability of degenerate parabolic equation with memory

“…Proof. For the proof see [3,Theorem 3.3], where this inequality is established for a coupled parabolic system in (t 0 , T ) × (0, 1) instead of Q. However, this inequality remains true in (0, T ) × (0, 1) with suitable changes.…”

Null controllability of degenerate parabolic equation with memory

“…Proof For the proof, see Allal et al, 22, Theorem 3.3 where this inequality is established for a coupled parabolic system in ( t 0 , T ) × (0, 1) instead of Q . However, this inequality remains true in (0, T ) × (0, 1) with suitable changes.…”