A uniqueness theorem for 3D semilinear wave equations satisfying the null condition

In this paper, we prove a uniqueness theorem for a system of semilinear wave equations satisfying the null condition in $\mathbb{R}^{1+3}$. Suppose that two global solutions with $C_c^\infty$ initial data have equal initial data outside a ball and equal radiation fields outside a light cone. We show that these two solutions are equal either outside a hyperboloid or everywhere in the spacetime, depending on the sizes of the ball and the light cone.Comment: 43 pages, 1 figure. Revised based on a referee repor

Data Dissemination in Unified Dynamic Wireless Networks

We give efficient algorithms for the fundamental problems of Broadcast and Local Broadcast in dynamic wireless networks. We propose a general model of communication which captures and includes both fading models (like SINR) and graph-based models (such as quasi unit disc graphs, bounded-independence graphs, and protocol model). The only requirement is that the nodes can be embedded in a bounded growth quasi-metric, which is the weakest condition known to ensure distributed operability. Both the nodes and the links of the network are dynamic: nodes can come and go, while the signal strength on links can go up or down. The results improve some of the known bounds even in the static setting, including an optimal algorithm for local broadcasting in the SINR model, which is additionally uniform (independent of network size). An essential component is a procedure for balancing contention, which has potentially wide applicability. The results illustrate the importance of carrier sensing, a stock feature of wireless nodes today, which we encapsulate in primitives to better explore its uses and usefulness.Comment: 28 pages, 2 figure

Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms

We study the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the B\"acklund transform which reduces the study of the asymptotic stability of the kinks to the study of the asymptotic decay of solutions near zero. Our results consist of two parts. First, we present a different proof of the local asymptotic stability result in arXiv:2009.04260. In its proof, we apply a result obtained by the inverse scattering method on the local decay of the solutions with sufficiently small and localized initial data. Moreover, we prove an $L^\infty$-type asymptotic stability result which is similar to that in arXiv:2106.09605; the main difference is that we remove the assumptions on the spatial symmetry of the perturbations. In its proof, we apply a result obtained by the method of testing by wave packets on the pointwise decay of the solutions with small and localized data.Comment: 54 page