Related papers: On the wave equation with a large rough potential

We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this…

We consider the non-monotone degenerate diffusion equation with time delay. Different from the linear diffusion equation, the degenerate equation allows for semi-compactly supported traveling waves. In particular, we discover…

We consider the initial value problem of the compressible Navier-Stokes-Korteweg equations in the whole space $\mathbb{R}^d$ ($d \ge 2$). The purposes of this paper are to obtain the global-in-time solution around the constant equilibrium…

Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on…

We establish Strichartz estimates (both reversed and some direct ones), pointwise decay estimates, and weighted decay estimates for the linear wave equation in dimension two with an almost scaling-critical potential, in the case when there…

We consider the Cauchy problem for the weakly dissipative wave equation $$ \square u+\frac\mu{1+t} u_t=0 $$ with parameter $\mu\ge2$. Based on the explicit representations of solutions provided in [Math. Meth. Appl. Sci. 2004; {\bf…

We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping…

This paper investigates the decay rates of the contact wave in one-dimensional Navier-Stokes equations. We study two cases of perturbations, with and without zero mass condition, i.e., the integration of initial perturbations is zero and…

We obtain weighted $L^2$ estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lam\'e operator…

We prove dispersive estimates at low frequency in dimensions n greater or equal to 4 for the wave equation for a very large class of real-valued potentials, provided the zero is neither an eigenvalue nor a resonance.

The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the…

For nonlinear wave equations with a potential term we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. We show that the perturbation series has a positive convergence radius by a method…

We prove weighted L^2 (Morawetz) estimates for the solutions of linear Schrodinger and wave equation with potentials that decay like |x|^{-2} for large x, by deducing them from estimates on the resolvent of the associated elliptic operator.…

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation…

Let \( H = (-\Delta)^m + V \) be a higher-order elliptic operator on \( L^2(\mathbb{R}^n) \), where \( V \) is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to…

Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a…

We prove sharp Strichartz-type estimates in three dimensions, including some which hold in reverse spacetime norms, for the wave equation with potential. These results are also tied to maximal operator estimates studied by…

A condition which guaranties the exponential decay of the solutions of the initial-boundary value problem for the damped wave equation is proved. A method for the effective computability of the coefficient of exponential decay is also…

In this paper we consider the local energy decay result for wave equations with a short-range potential. It is important to note that one never uses a finite speed of propagation property unlike the historical previous papers. The essential…

We study the hyperboloidal initial value problem for the one-dimensional wave equation perturbed by a smooth potential. We show that the evolution decomposes into a finite-dimensional spectral part and an infinite-dimensional radiation…