Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms (Q1608137)

\noindent The system is described by the parabolic equation \[ \begin{align*}{ {\partial y(t, x) \over \partial t} &= \Delta y(t, x) - f(y(t, x), \nabla y(t, x)) + u(t, x) \chi_\omega(x) \quad (t \ge 0, x \in \Omega) , \cr y(t, x) &= 0 \quad (t \ge 0, x \in \Gamma) , \qquad u(0, x) = u_0(x) \quad (x \in \Omega) }\end{align*} \] in an unbounded \(n\)-dimensional domain \(\Omega\) with boundary \(\Gamma;\) \(\chi_\omega(x)\) is the characteristic function of a subdomain \(\omega \subset \Omega\) and \(u(\cdot, \cdot) \in L^2((0, T) \times \Omega)\) is the control. The nonlinearity satisfies \(f(0, 0) = 0.\) The authors show that, if \(\Omega \setminus \omega\) is bounded, the system is null controllable under suitable smoothness assumptions.