scholarly journals Weighted energy estimates for wave equations in exterior domains

2011 ◽  
Vol 23 (6) ◽  
Author(s):  
Mishio Kawashita ◽  
Hiroshi Sugimoto
Keyword(s):  
Wave Equations ◽  
Energy Estimates ◽  
Exterior Domains ◽  
Weighted Energy ◽  
Weighted Energy Estimates

scholarly journals The lifespan for 3-dimensional quasilinear wave equations in exterior domains

2014 ◽  
Vol 26 (6) ◽  
Author(s):  
John Helms ◽  
Jason Metcalfe
Keyword(s):  
Wave Equations ◽  
Energy Estimates ◽  
Exterior Domains ◽  
Small Initial Data ◽  
3 Dimensional ◽  
Long Time Existence ◽  
Homogeneous Wave ◽  
Long Time ◽  
Quasilinear Wave Equations ◽  
Time Existence

AbstractThis article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

scholarly journals Convergence rates to the damped system of compressible adiabatic flow through porous media with boundary effect

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Lina Zhang ◽  
Shifeng Geng ◽  
Yuling Gao
Keyword(s):  
Porous Media ◽  
Convergence Rates ◽  
Boundary Effect ◽  
Energy Estimates ◽  
Flow Through Porous Media ◽  
Adiabatic Flow ◽  
Weighted Energy ◽  
Damped System ◽  
Weighted Energy Estimates ◽  
Flow Through

AbstractIn this paper, we consider convergence rates to solutions for the damped system of compressible adiabatic flow through porous media with boundary effect. Compared with the results obtained by Pan, the better convergence rates are obtained in this paper. Our approach is based on the technical time-weighted energy estimates.

scholarly journals Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data

2019 ◽  
Vol 21 (05) ◽  
pp. 1850035
Author(s):  
Motohiro Sobajima ◽  
Yuta Wakasugi
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Hypergeometric Functions ◽  
Smooth Boundary ◽  
Decay Rates ◽  
Energy Estimates ◽  
Damping Term ◽  
Confluent Hypergeometric Functions ◽  
Weighted Energy ◽  
Weighted Energy Estimates

This paper is concerned with weighted energy estimates for solutions to wave equation [Formula: see text] with space-dependent damping term [Formula: see text] [Formula: see text] in an exterior domain [Formula: see text] having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in [Formula: see text]. The crucial idea is to use special solution of [Formula: see text] including Kummer’s confluent hypergeometric functions.

scholarly journals Global existence of null-form wave equations in exterior domains

2007 ◽  
Vol 256 (3) ◽  
pp. 521-549 ◽  
Author(s):  
Jason Metcalfe ◽  
Christopher D. Sogge
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Exterior Domains

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

An inverse scattering theorem for (1 + 1)-dimensional semi-linear wave equations with null conditions

2021 ◽  
Vol 18 (01) ◽  
pp. 143-167
Author(s):  
Mengni Li
Keyword(s):  
Inverse Scattering ◽  
Wave Equations ◽  
Scattering Problem ◽  
Weighted Sobolev Spaces ◽  
One Dimension ◽  
Energy Estimates ◽  
Small Data ◽  
Linear Wave ◽  
Global Existence Of Solutions ◽  
Linear Wave Equations

We are interested in the inverse scattering problem for semi-linear wave equations in one dimension. Assuming null conditions, we prove that small data lead to global existence of solutions to [Formula: see text]-dimensional semi-linear wave equations. This result allows us to construct the scattering fields and their corresponding weighted Sobolev spaces at the infinities. Finally, we prove that the scattering operator not only describes the scattering behavior of the solution but also uniquely determines the solution. The key ingredient of our proof is the same strategy proposed by Le Floch and LeFloch [On the global evolution of self-gravitating matter. Nonlinear interactions in Gowdy symmetry, Arch. Ration. Mech. Anal. 233 (2019) 45–86] as well as Luli et al. [On one-dimension semi-linear wave equations with null conditions, Adv. Math. 329 (2018) 174–188] to make full use of the null structure and the weighted energy estimates.

Sign in / Sign up

Export Citation Format

Share Document