Observability and null-controllability for parabolic equations in $ L_p $-spaces

We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) x_0\in L_p (\mathbb{R}^d)$, with interior a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, $A$ is an elliptic operator order $m \in \mathbb{N}$ $L_p(\mathbb{R}^d)$. prove this system via duality sufficient condition for observability. This given by uncertainty principle dissipation estimate. Our result unifies generalizes earlier results obtained context Hilbert Banach spaces. particular, our applies to case $p=1$.