On the Bishop-Phelps-Bollob\'as property for numerical radius

We study the Bishop-Phelps-Bollob\'as property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces ensuring the BPBp-nu. Among other results, we show that $L_1(\mu)$-spaces have this property for every measure $\mu$. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikod\'{y}m property (even reflexivity) is not enough to get BPBp-nu.