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-ℓ(Ω)), ℓ ∈ [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.
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Masahiro Yamamoto, Oleg Yu. Emanouilov (Imanuvilov), Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for Semilinear Parabolic Equations. Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, pp. 227–274DOI 10.2977/PRIMS/1145476103