Advances in Differential Equations

Some results on controllability for linear and nonlinear heat equations in unbounded domains

Manuel González-Burgos and Luz de Teresa

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Abstract

In this paper we present some results concerning the null controllability for a heat equation in unbounded domains. We characterize the conditions that must satisfy the auxiliary function that leads to a global Carleman inequality for the adjoint problem and then to get a null controllability result. We give some examples of unbounded domains $(\Omega,\omega)$ that satisfy these sufficient conditions. Finally, when $\Omega\setminus \overline \omega$ is bounded, we prove the null controllability of the semilinear heat equation when the nonlinearity $f(y,\nabla y)$ grows more slowly than $|y|\log^{3/2}(1+|y|+|\nabla y|)+ |\nabla y|\log^{1/2}(1+|y|+|\nabla y|)$ at infinity (generally in this case in the absence of control, blow-up occurs). In this aim we also prove the linear null controllability problem with $L^\infty$-controls.

Article information

Source
Adv. Differential Equations Volume 12, Number 11 (2007), 1201-1240.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867413

Mathematical Reviews number (MathSciNet)
MR2372238

Zentralblatt MATH identifier
1170.93007

Subjects
Primary: 93B05: Controllability
Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 93B07: Observability 93C20: Systems governed by partial differential equations

Citation

González-Burgos, Manuel; de Teresa, Luz. Some results on controllability for linear and nonlinear heat equations in unbounded domains. Adv. Differential Equations 12 (2007), no. 11, 1201--1240. https://projecteuclid.org/euclid.ade/1355867413.


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