29 citations · 174 across the 25 of their papers we have counts for
26 papers
Observability for wave equations with critically singular potentials
Vaibhav Kumar Jena, Arick Shao
In this article, we prove boundary observability estimates for wave equations on bounded domains , with a critically singular potential that diverges as th…
Controllability of a One-Dimensional Dynamic Debonding Model
Nicola De Nitti, Arick Shao
We investigate a one-dimensional dynamic debonding model, introduced by Freund (1990), in which the wave equation is coupled with a Griffith criterion governing the propagation of…
Asymptotics and Scattering for Critically Weakly Hyperbolic and Singular Systems
Bolys Sabitbek, Arick Shao
We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain lower-order coefficients that blow up at a single time …
Approximate boundary controllability for parabolic equations with inverse square infinite potential wells
Arick Shao, Bruno Vergara
We consider heat operators on a bounded domain , with a critically singular potential diverging as the inverse square of the distance to . Whil…
On counterexamples to unique continuation for critically singular wave equations
Simon Guisset, Arick Shao
We consider wave equations with a critically singular potential diverging as an inverse square at a hypersurface . Our aim is to construct counterexamples to…
Bulk-boundary correspondences and unique continuation in asymptotically Anti-de Sitter spacetimes
Arick Shao
This article surveys the research presented by the author at the MATRIX Institute workshop "Hyperbolic Differential Equations in Geometry and Physics" in April 2022. The work is ce…
The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes
Gustav Holzegel, Arick Shao
In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes with conformal boundary . We establish a corresponden…
A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes
Athanasios Chatzikaleas, Arick Shao
We reconsider the unique continuation property for a general class of tensorial Klein-Gordon equations of the form \begin{align*} \Box_{g} ϕ+ σϕ= \mathcal{G}(ϕ,\nabla ϕ) \text{,} \…
Control of waves on Lorentzian manifolds with curvature bounds
Vaibhav Kumar Jena, Arick Shao
We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The…
Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates
Alberto Enciso, Arick Shao, Bruno Vergara
We consider heat operators on a convex domain , with a critically singular potential that diverges as the inverse square of the distance to the boundary of . We establish a g…
Null Geodesics and Improved Unique Continuation for Waves in Asymptotically Anti-de Sitter Spacetimes
Alex McGill, Arick Shao
We consider the question of whether solutions of Klein--Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive…
Global stability of traveling waves for -dimensional systems of quasilinear wave equations
Louis Dongbing Cha, Arick Shao
A key feature of -dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In thi…
The Near-Boundary Geometry of Einstein-Vacuum Asymptotically Anti-de Sitter Spacetimes
Arick Shao
We study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite…
Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials
Alberto Enciso, Arick Shao, Bruno Vergara
We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse s…
On Carleman and Observability Estimates for Wave Equations on Time-Dependent Domains
Arick Shao
We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main fea…
Unique continuation from infinity in asympotically Anti-de Sitter spacetimes II: Non-static boundaries
Gustav Holzegel, Arick Shao
We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations on asymptotically…
Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes
Gustav Holzegel, Arick Shao
We consider the unique continuation properties of asymptotically Anti-de Sitter spacetimes by studying Klein-Gordon-type equations , $σ\…
On the profile of energy concentration at blow-up points for sub-conformal focusing nonlinear waves
Spyros Alexakis, Arick Shao
We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaa…
Global uniqueness theorems for linear and nonlinear waves
Spyros Alexakis, Arick Shao
We prove a unique continuation from infinity theorem for regular waves of the form . Under the assumption of no incoming and no outgoing radiation…
Unique continuation from infinity for linear waves
Spyros Alexakis, Volker Schlue, Arick Shao
We prove various uniqueness results from null infinity, for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts…
Bounds on the Bondi Energy by a Flux of Curvature
Spyros Alexakis, Arick Shao
We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild sp…
On the Geometry of Null Cones to Infinity Under Curvature Flux Bounds
Spyros Alexakis, Arick Shao
The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particul…
Hamiltonian dynamics of a particle interacting with a wave field
Daniel Egli, Jürg Fröhlich, Zhou Gang +2
We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. So…
New tensorial estimates in Besov spaces for time-dependent -dimensional problems
Arick Shao
In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this…