Scattering for radial energy-subcritical wave equations in dimensions 4 and 5

Abstract

In this paper, we consider the focusing and defocusing energy-subcritical, nonlinear wave equation in R 1+d with radial initial data for d = 4, 5. We prove that if a solution remains bounded in the critical space on its interval of existence, then the solution exists globally and scatters at ±∞. The proof follows the concentration compactness/rigidity method initiated by Kenig and Merle, and the main obstacle is to show the nonexistence of nonzero solutions with a certain compactness property. A main novelty of this work is the use of a simple virial argument to rule out the existence of nonzero solutions with this compactness property rather than channels of energy arguments that have been proven to be most useful in odd dimensions. 4

Soliton Resolution for Equivariant Wave Maps on a Wormhole

“…These two steps are carried out in Section 3. We remark here that in the work [13] the authors established these steps by using delicate estimates and arguments developed in [3] and [18] for the energy-critical wave equation on flat space in high dimensions. This is done by using a Strauss estimate to reduce the nonlinearity to an energy-critical power on R 1+(2ℓ+3) .…”

Universality of Blow up Profile for Small Blow up Solutions to the Energy Critical Wave Map Equation

“…The new ingredient in these works is the channel of energy inequality first introduced in [20,21], which provides strong decoupling mechanism between the dispersion and solitary waves. The channel of energy type inequality has been applied to many other problems, see for example [12,24,38,39,58] for application to semilinear energy critical wave equations. These channel of energy inequalities in many cases depend crucially on the radial assumption and are sensitive to the dimensions.…”

A sharp Lorentz-invariant Strichartz norm expansion for the cubic wave equation in $\mathbb{R}^{1+3}$

“…H 1{2 ; see [3,4,7,21,22] and the recent [5,6,8]. However, to the knowledge of the author, the only paper, other than the present one, that deals with Lorentzian transformations is the work of Ramos [20]; see also [17] for the Klein-Gordon equation.…”