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Consider the focusing 4D cubic wave equation [ partial_{tt}u-Delta u-u^{3}=0,quad mbox{on} (t,x)in [0,infty)times mathbb{R}^{4}.] The main result states the existence in energy space $dot{H}^{1}times L^{2}$ of multi-solitary waves where each traveling wave is generated by Lorentz transform from a specific excited state, with different but collinear Lorentz speeds. The specific excited state is deduced from the non-degenerate sign-changing state constructed in Musso-Wei [34]. The proof is inspired by the techniques developed for the 5D energy-critical wave equation and the nonlinear Klein-Gordon equation in a similar context by Martel-Merle [30] and C^ote-Martel [6]. The main difficulty originates from the strong interactions between solutions in the 4D case compared to other dispersive and wave-type models. To overcome the difficulty, a sharp understanding of the asymptotic behavior of the excited states involved and of the kernel of their linearized operator is needed.
For the 5D energy-critical wave equation, we construct excited $N$-solitons with collinear speeds, i.e. solutions $u$ of the equation such that begin{equation*} lim_{tto+infty}bigg| abla_{t,x}u(t)- abla_{t,x}bigg(sum_{n=1}^{N}Q_{n}(t)bigg)bigg|_{
Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded o
We consider the hyperboloidal initial value problem for the cubic focusing wave equation. Without symmetry assumptions, we prove the existence of a co-dimension 4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves.
In this paper, we construct $K$-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions $u$ of the equation such that begin{equation*} | abla_{t,x}u(t)- abla_{t,x}big(sum_{k=1}^{K}W_{k}(t)big)|_{L^{2
We consider the initial value problem for the spherically symmetric, focusing cubic wave equation in three spatial dimensions. We give numerical and analytical evidence for the existence of a universal attractor which encompasses both global and blow