Null controllability for semilinear heat equation with dynamic boundary conditions

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ F $\end{document}</tex-math></inline-formula> that behaves at infinity like <inline-formula><tex-math id="M2">\begin{document}$ |s| \ln ^{p}(1+|s|), $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ p &gt; 2 $\end{document}</tex-math></inline-formula>. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties <inline-formula><tex-math id="M4">\begin{document}$ F(s, \xi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ G(s, \xi) $\end{document}</tex-math></inline-formula> growing slower than <inline-formula><tex-math id="M6">\begin{document}$ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $\end{document}</tex-math></inline-formula> at infinity.</p>