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

Source 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^{-ℓ}(Ω)) , ℓ \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.