Carleman inequalities and the heat operator
The paper refers to the strong unique continuation property for parabolic equations with time-dependent coefficients. The strong unique continuity result for the heat operator is derived from a Carleman inequality: \[ \|t^{-\varepsilon} e^{-|x|^{2/8t}}f\|_{L^q_tL^p_x}\leq N(n,r,s,\mu)\|t^{-\alpha+1-(n/2r)-(1/s)}e^{-|x|^2/8t}(\Delta f+\partial_tf)\|_{L^p_t L^p_x} \] which holds under special assumptions for the parameters and for any \(f\in C^\infty_0(\mathbb{R}^{n+1}\setminus\{(0,0)\})\). This estimation is proved first and is actually the central result of the paper.
parabolic equations with time-dependent coefficients
scientific article; zbMATH DE number 1501667
scientific article