In this Letter, we introduce the nonlinear partial differential equation ${\partial }_{\tau }^2\pi \propto (\overrightarrow{\nabla }\pi {)}^2$ showing a new type of instability. Such equations appear in the effective field theory (EFT) of dark energy for the $k$-essence model as well as in many other theories based on the EFT formalism. We demonstrate the occurrence of instability in the cosmological context using a relativistic $N$-body code, and we study it mathematically in $3+1$ dimensions within spherical symmetry. We show that this term dominates for the low speed of sound limit where some important linear terms are suppressed.

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• Alternative gravity theories
• Dark energy
• Evolution of the Universe
• Large scale structure of the Universe