Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms

We consider the null controllability problem for the semilinear heat equation with nonlinearities involving gradient terms in an unbounded domain of ^N with Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain such that the uncontrolled region is bounded. Using Carleman inequalities, we prove first the null controllability of the linearized equation. Then, by a fixed-point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is ^C1 and globally Lipschitz, the system is null controllable.