Nondispersive decay for the cubic wave equation

Abstract

We consider the hyperboloidal initial value problem for the cubic focusing wave equation (-∂^2_t+Δ_x)v(t,x)+v(t,x)^3 = 0, x Є ℝ^3. Without symmetry assumptions, we prove the existence of a codimension-4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves. More precisely, for any δ∈(0,1), we construct solutions with the asymptotic behavior ║v-v_0‖_(L^4(t2t)L^4(B_((1-δ)t))≾ t^(-1/2+) as t → ∞, where v_0(t,x)= √2/t and B_((1-δ)t):={x Є ℝ^3:|x|<(1-δ)t}.

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