𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

✍ Scribed by O. Bodart; M. González-Burgos; R. Pérez-Garcı́a

Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
366 KB
Volume
57
Category
Article
ISSN
0362-546X

No coin nor oath required. For personal study only.

✦ Synopsis

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of R N . We ÿrst consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F(y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L 2 -norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate ÿxed point arguments.

📜 SIMILAR VOLUMES

Insensitizing controls for a nonlinear p
Insensitizing controls for a nonlinear parabolic equation with a nonlinear term involving the state and the gradient in unbounded domains
✍ Youjun Xu; Xinguang Zhang; Dongyuan Liu 📂 Article 📅 2009 🏛 Elsevier Science 🌐 English ⚖ 600 KB

In this paper, we consider the existence of insensitizing control for a semilinear heat equation involving gradient terms in unbounded domain Ω. In this case, we prove the existence of controls insensitizing the L 2 -norm of the observation of the solution in an open subset of the domain. The proofs

A Liouville theorem and blowup behavior
A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure
✍ Hatem Zaag 📂 Article 📅 2000 🏛 John Wiley and Sons 🌐 English ⚖ 178 KB 👁 1 views

We prove a Liouville theorem for the following heat system whose nonlinearity has no gradient structure: where pq > 1, p ≥ 1, q ≥ 1, and | p -q| small. We then deduce a localization property and uniform L ∞ estimates of blowup solutions of this system.

A singular nonlinear elliptic equation w
A singular nonlinear elliptic equation with natural growth in the gradient
✍ Wen-Shu Zhou 📂 Article 📅 2008 🏛 John Wiley and Sons 🌐 English ⚖ 150 KB

## Abstract This paper is devoted to the existence and regularity of the homogenous Dirichlet boundary value problem for a singular nonlinear elliptic equation with natural growth in the gradient. By certain transformations, the problem can be transformed formally into either a Dirichlet problem or