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We study the asymptotic behaviour, as time goes to infinity, of theFisher-KPP equation $\partial_t u=\Delta u +u-u^2$ in spatial dimension $2$,when the initial condition looks like a Heaviside function. Thus the solutionis, asymptotically in time, trapped between two planar critical waves whosepositions are corrected by the Bramson logarithmic shift. The issue is whether,in this reference frame, the solutions will converge to a travelling wave, orwill exhibit more complex behaviours. We prove here that both convergence andnonconvergence may happen: the solution may converge towards one translate ofthe planar wave, or oscillate between two of its translates. This relies on thebehaviour of the initial condition at infinity in the transverse direction.

Nontrivial dynamics beyond the logarithmic shift in two-dimensional Fisher-KPP equations