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We consider the quadratic semilinear wave equation in six dimensions. Thisenergy critical problem admits a ground state solution, which is the unique (upto scaling) positive stationary solution. We prove that any sphericallysymmetric solution, that remains bounded in the energy norm, evolvesasymptotically to a sum of decoupled modulated ground states, plus a radiationterm. As a by-product of the approach we prove the non-existence ofmultisoliton solutions that do not emit any radiation. The proof follows themethod initiated for large odd dimensions by the last three authors, reducingthe problem to ruling out the existence of such non-radiative multisolitons, byderiving a contradiction from a finite dimensional system of ordinarydifferential equations governing their modulation parameters. In comparison,the difficulty in six dimensions is the failure of certain channel of energyestimates and the related existence of a linear resonance. The main noveltiesare the use of new channel of energy estimates, as well as the classificationof non-radiative solutions with small energy.

Soliton resolution for the radial quadratic wave equation in six space dimensions