We consider the 2D cubic Zakharov–Kuznetsov (ZK) equation, a physically relevant model in physics, which is a higher-dimensional extension of the generalized KdV equation. The cubic ZK equation is \(L^2\) -critical in 2D and exhibits instability of solitons as we have shown in [10]. Here we prove that near-threshold negative energy solutions to this ZK equation blow-up in finite or infinite time, the first such result for higher-dimensional extensions of the gKdV family of equations. The proof consists of several steps. First, we show that if the blow-up conclusion is false, there are negative energy solutions arbitrarily close to the threshold that are globally bounded in \(H^1\) and are spatially localized, uniformly in time. In the second step, we show that such solutions must in fact be exact remodulations of the ground state, and hence, have zero energy, which is a contradiction. This second step, a nonlinear Liouville theorem, is proved by contradiction, with a limiting argument producing a nontrivial solution to a (linear) linearized ZK equation obeying uniform-in-time spatial localization. Such nontrivial linear solutions are excluded by a local-virial space-time estimate. We introduce several new features to handle the 2D cubic ZK equation.
