Construction of infinite time bubble tower solutions to critical wave maps equation

Abstract

We construct infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. More precisely, for any integers $k\geq3$ and $J\geq1$, we construct a solution that is global in one time direction, has $k$-corotational symmetry, and asymptotically decomposes into $J$-many concentric bubbles of alternating signs with asymptotically vanishing radiation. The scales of each bubble are of order $t^{-\alpha_{j}}$ with $\alpha_{j}=(\frac{k}{k-2})^{j-1}-1$. This shows the existence of multi-bubble solutions with an arbitrary number of bubbles in soliton resolution, provided that $k\geq3$, global existence in one time direction, and alternating signs are considered. Our proof is based on modulation analysis with the method of backward construction. The key new ingredient is a Morawetz-type functional that provides suitable monotonicity estimates for solutions around multi-bubble configurations.