A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

¹ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.
² Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China.

^* Corresponding author: zhangcansx@163.com

Received: 11 May 2021
Accepted: 4 January 2022

Abstract

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space ℝ^N. As an application, we then show the exact null-controllability for this semilinear heat equation in ℝ^N. The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in ℝ^N. This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.