Related papers: Pointwise convergence for the elastic wave equatio…
Our goal is to establish existence with suitable initial data of solutions to general parabolic equation in one dimension, $u_t = L(u_x)_x$, where $L$ is merely a monotone function. We also expose the basic properties of solutions,…
The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12]…
For the Cauchy problem of nonlinear elastic wave equations of three dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in…
The paper concerns $L^1$- convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials $(-4\le \gm<0$), with and without angular cutoff. We prove the time-averaged $L^1$-convergence to…
In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space…
The peridynamic analogue of the wave equation does not have finite speed propogation. We show, for one dimensional linear peridynamics, that solutions do nonetheless satisfy estimates analogous to those satisfied by solutions of the wave…
We study the wave front set of the solutions of the initial value problem for nonlinear Schr\"{o}dinger equations via wave packet transform. We give an sufficient condition which assures that the solutions is in Sobolev space of order s in…
We prove that the initial-value problem for the motion of a certain type of elastic body has a solution for all time if the initial data are sufficiently small. The body must fill all of three space, obey a ``neo-Hookean'' stress-strain…
We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate $1/t$. We also establish observability results, at…
In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…
We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our…
We consider a semilinear wave equation in the whole space with a deep potential well. We prove that as the depth of the well tends to infinity, the solutions of the equation converge to the solutions of a wave equation defined on the bottom…
In the paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the new technical which is developed in \cite{Li2}, we proved that the data-to-solution map of this problem is not…
This article represents a first step toward understanding the long-time dynamics of solutions for the Intermediate Long Wave equation (ILW). While this problem is known to be both completely integrable and globally well-posed in…
Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately…
We consider linear and non-linear Cauchy equations in the context of Sobolev spaces. In particular, we show the global existence of solutions to the Kirchhoff equation with initial data in the Sobolev spaces, a problem that has been open…
We study a 1D semilinear wave equation modeling the dynamic of an elastic string interacting with a rigid substrate through an adhesive layer. The constitutive law of the adhesive material is assumed elastic up to a finite critical state,…
In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the…
The wavefronts from a point source in a solid with cubic symmetry are examined with particular attention paid to the contribution from the conical points of the slowness surface. An asymptotic solution is developed that is uniform across…
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate,…