Unique continuation inequalities for the parabolic-elliptic chemotaxis system

The paper proves the two one-time unique continuation inequalities for the parabolic–elliptic chemotaxis system by (i) turning the u-equation into a linear parabolic equation with bounded lower-order terms after establishing L∞ bounds on v and ∇v, (ii) deriving a local frequency-function-based interpolation inequality and propagating it to ω via localization, and (iii) obtaining a refined H−1-based estimate to recover the initial data; see Theorem 1.1 (inequalities (1.2)–(1.3)) and the proof outline given by the authors themselves . The candidate solution follows the same blueprint: elliptic regularity and L∞ a priori bounds to control the coefficients, frequency-function arguments for a one-time interpolation, an elliptic resolvent estimate to incorporate v(T), and an H−1 frequency parameter to obtain the logarithmic estimate. Minor differences concern technical localization (chain-of-balls) and the stated dependence of constants; otherwise the approaches are essentially the same and correct.