Null controllability for semilinear heat equation with dynamic boundary conditions

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 \begin{document}$ F $\end{document} that behaves at infinity like \begin{document}$ |s| \ln ^{p}(1+|s|), $\end{document} with \begin{document}$ p > 2 $\end{document} . Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties \begin{document}$ F(s, \xi) $\end{document} and \begin{document}$ G(s, \xi) $\end{document} growing slower than \begin{document}$ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $\end{document} at infinity.