The aim of this paper is to study the wellposedness and L 2 -regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space.2010 Mathematics Subject Classification. 35A15; 35K20; 47D06.

The aim of this paper is to study the wellposedness and L2‐regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space.

<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>

<p style='text-indent:20px;'>In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term.</p>