Related papers: Price's Law on Nonstationary Spacetimes
We investigate one-dimensional scalar balance laws with singular convolution-type source terms. Under appropriate convexity and kernel assumptions, we establish the global existence of entropy weak solutions in ${\bf L}^2(\mathbb{R})$,…
We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schr\"odinger equation in spatial dimensions $d = 3,4$ for both the initial-value and final-state problems.
We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized…
In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with a nonlinearly degenerate viscosity where the far field states are prescribed. Especially, we deal with…
We consider the total energy decay together with L^2-bound of the solution itself of the Cauchy problem for wave equations with a localized damping and a short-range potential. We treat it in the one dimensional Euclidean space R. We adopt…
It is classical that uniform stabilization of solutions to the autonomous damped wave equation is equivalent to every geodesic meeting the positive set of the damping, which is called the geometric control condition. In this paper, it is…
We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation…
We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also…
We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter spacetimes, and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our…
Supersymmetric microstate geometries were recently conjectured to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
We consider the Cauchy problem for wave equations with variable coefficients in the whole space. We improve the rate of decay of the local energy, which has been recently studied by J. Shapiro, where he derives the log-order decay rates of…
We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global…
We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension 1 in presence of a flat bottom. We prove a decay with respect to time t of order 1/3 for solutions with initial data in weighted Sobolev…
We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From…
We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrodinger equation; second, the notion of full asymptotic stability (which states that…
We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an…
Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant…
We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial…
In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ${\mathbb R} \times {\mathbb R}^d$ with $d \geq 6$. We prove the stability of solutions under the weak condition…