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 of the 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 control on a so-called thick set $E \subset \mathbb{ R}^d$, where $p\in [1,\infty)$, and where $A$ is an elliptic operator of order $m \in \mathbb{ N}$ in $L_p(\mathbb{ R}^d)$. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case $p=1$.
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