Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for Semilinear Parabolic Equations

Abstract

We prove Carleman inequalities for a second order parabolic equation when the coeffcients are not bounded and norms of right hand sides are taken in the Sobolev space $L^2(0,T;W_2^{-\ell }(\Omega ))$, $\ell \in [0,1]$. Our Carleman inequality yields the unique continuation for $L^2$-solutions. We further apply these inequalities to the global exact zero controllability of a semilinear parabolic equation whose semilinear term also contains derivatives of first order of solutions and is of sub-linear growth at the infinity.