Hadamard ill-posedness of the linearised Prandtl equations in Gevrey spaces

We prove Hadamard ill-posedness for the non-autonomous linearised Prandtl equations around time-dependent shear-flow equilibria in function spaces up to Gevrey class 4. More precisely, we construct compactly supported smooth initial data, Gevrey class 4 in the tangential variable, for which the system admits no weak solution for any positive lifespan. In this regard, we improve previous results by showing that classical semigroup-type instabilities do not, by themselves, imply Hadamard ill-posedness in the non-autonomous case when the initial time is not a variable of the system. Our argument is based on a family of exact unstable modes whose $L^2$-norms grow, at tangential frequency $k$, like $\exp(c \sqrt k t)$ up to times of order $t \sim k^{-1/4}$. Their construction relies on an inner-outer gluing scheme, in the spirit of matched asymptotic expansions, which combines unstable inner solutions near a non-degenerate critical point with an exact outer solution and yields exponentially small matching errors for short times.