In this paper we present some results concerning the null controllability for a heat equation in unbounded domains. We characterize the conditions that must satisfy the auxiliary function that leads to a global Carleman inequality for the adjoint problem and then to get a null controllability result. We give some examples of unbounded domains $(\Omega,\omega)$ that satisfy these sufficient conditions. Finally, when $\Omega\setminus \overline \omega$ is bounded, we prove the null controllability of the semilinear heat equation when the nonlinearity $f(y,\nabla y)$ grows more slowly than $|y|\log^{3/2}(1+|y|+|\nabla y|)+ |\nabla y|\log^{1/2}(1+|y|+|\nabla y|)$ at infinity (generally in this case in the absence of control, blow-up occurs). In this aim we also prove the linear null controllability problem with $L^\infty$-controls.

We consider a model of non-isothermal phase transition taking place in a confined container. The order parameter $\phi $ is governed by a Cahn--Hilliard-type equation which is coupled with a heat equation for the temperature $\theta $. The former is subject to a nonlinear dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or even dynamic. We thus formulate a class of initial- and boundary-value problems whose local existence and uniqueness is proven by means of the contraction mapping principle. The local solution becomes global owing to suitable a priori estimates.

Let $X$ be a real reflexive Banach space with dual $X^*.$ Let $T:X\supset D(T)\to 2^{X^*}$ be maximal monotone, and $C:X\supset D(C)\to X^*.$ A theory of domain invariance is developed, in which various conditions are given for a nonlinear operator of the type $T+C:D(T)\cap D(C)\to 2^{X^*}$ to map a given relatively open set onto an open set. The well-known invariance of domain theorem of Schauder about injective operators of the type $I+C,$ with $C$ compact, is extended to operators $T+C.$ Here, $T$ is a possibly densely defined operator with compact resolvents and $C$ is continuous and bounded, or $T$ is just maximal monotone and $C$ compact. The case of completely continuous resolvents of $T$ and demicontinuous operators $C$ is also covered. Another invariance of domain result is given for demicontinuous, bounded, and $(S_+)$-perturbations $C.$ This result makes use of the topological degrees of Browder and Skrypnik. Finally, three invariance of domain theorems are given for the case where $T$ is single-valued and both operators $T,~C$ are densely defined. These results make use of the topological degree theory that was recently developed by the authors for the sum $T+C,$ where $C$ satisfies conditions like quasiboundedness and $(S_+)$ with respect to $T.$ Applications to elliptic and parabolic problems are included.