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

We study the asymptotic behaviour, as time goes to infinity, of the Fisher-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 solution is, asymptotically in time, trapped between two planar critical waves whose positions are corrected by the Bramson logarithmic shift. The issue is whether, in this reference frame, the solutions will converge to a travelling wave, or will exhibit more complex behaviours. We prove here that both convergence and nonconvergence may happen: the solution may converge towards one translate of the planar wave, or oscillate between two of its translates. This relies on the behaviour of the initial condition at infinity in the transverse direction.