Null controllability for semilinear heat equation with dynamic boundary conditions

Abdelaziz Khoutaibi; Lahcen Maniar; Omar Oukdach

doi:10.3934/dcdss.2022087

Article Contents

2022, Volume 15, Issue 6: 1525-1546. Doi: 10.3934/dcdss.2022087

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Null controllability for semilinear heat equation with dynamic boundary conditions

Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), B.P. 2390, Marrakesh, Morocco

^* Corresponding author: Omar Oukdach
^* Corresponding author: Omar Oukdach

Received: December 2021

Revised: March 2022

Published: June 2022

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Abstract

This paper deals with the null controllability of the semilinear heat equation with dynamic boundary conditions of surface diffusion type, with nonlinearities involving drift terms. First, we prove a negative result for some function $ F $ that behaves at infinity like $ |s| \ln ^{p}(1+|s|), $ with $ p > 2 $. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties $ F(s, \xi) $ and $ G(s, \xi) $ growing slower than $ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $ at infinity.

Keywords:

Null controllability,
observability,
semilinear heat equation with dynamic boundary conditions.

Mathematics Subject Classification: Primary: 93Bxx; Secondary: 35K05.

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