Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations

We prove Carleman inequalities for a second order parabolic equation when the coefficients are not bounded and norms of right hand sides are taken in the Sobolev space L-2 (0, T; W-2(-l) (Omega)), l epsilon [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.