Journal article

Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation (−∂2t+Δ)u+u5=0 in R1+3 and prove the existence of energy class solutions which are of the form u(t,x)=tμ2W(tμx)+η(t,x) in the forward lightcone {(t,x)∈R×R3:|x|≤t,t≫1} where W(x)=(1+13|x|2)−12 is the ground state soliton, μ is an arbitrary prescribed real number (positive or negative) with |μ|≪1 , and the error η satisfies ∥∂tη(t,⋅)∥L2(Bt)+∥∇η(t,⋅)∥L2(Bt)≪1,Bt:={x∈R3:|x|<t} for all t≫1. Furthermore, the kinetic energy of u outside the cone is small. Consequently, depending on the sign of μ, we obtain two new types of solutions which either concentrate as t→∞ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.

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