scholarly journals Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

2016 ◽  
Vol 2 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Motohiro Sobajima ◽  
◽  
Yuta Wakasugi
Keyword(s):  
Elliptic Problem ◽  
Exterior Domain ◽  
Wave Equations ◽  
Energy Estimates ◽  
Damping Term ◽  
Weighted Energy ◽  
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 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 Regularity of solutions to space–time fractional wave equations: A PDE approach

2018 ◽  
Vol 21 (5) ◽  
pp. 1262-1293 ◽  
Author(s):  
Enrique Otárola ◽  
Abner J. Salgado
Keyword(s):  
Elliptic Problem ◽  
Wave Equations ◽  
Time Derivative ◽  
Regularity Of Solutions ◽  
Energy Estimates ◽  
Infinite Cylinder ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Wave Equations ◽  
Fractional Time Derivative

Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

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.

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