scholarly journals STRICHARTZ ESTIMATES FOR WAVE EQUATION WITH INVERSE SQUARE POTENTIAL

2013 ◽  
Vol 15 (06) ◽  
pp. 1350026 ◽  
Author(s):  
CHANGXING MIAO ◽  
JUNYONG ZHANG ◽  
JIQIANG ZHENG
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Strichartz Estimates ◽  
Linear Wave ◽  
Well Posedness ◽  
Linear Wave Equation ◽  
Admissible Pairs

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential [Formula: see text] for power p being in some regime when the initial data are radial. This result extends the well-posedness result in Planchon, Stalker, and Tahvildar-Zadeh.

ON SELF-SIMILAR SOLUTIONS, WELL-POSEDNESS AND THE CONFORMAL WAVE EQUATION

2002 ◽  
Vol 04 (02) ◽  
pp. 211-222 ◽  
Author(s):  
FABRICE PLANCHON
Keyword(s):  
Wave Equation ◽  
Linear Wave ◽  
Well Posedness ◽  
Initial Value ◽  
Linear Wave Equation ◽  
Conformally Invariant ◽  
Self Similar ◽  
Similar Solutions ◽  
Well Posed ◽  
Homogeneous Data

We prove that the initial value problem for the conformally invariant semi-linear wave equation is well-posed in the Besov space [Formula: see text]. This induces the existence of (non-radially symmetric) self-similar solutions for homogeneous data in such Besov spaces.

Global well-posedness and linearization of a semilinear 2D wave equation with exponential type nonlinearity

2010 ◽  
Vol 17 (3) ◽  
pp. 543-562 ◽  
Author(s):  
Olfa Mahouachi ◽  
Tarek Saanouni
Keyword(s):  
Wave Equation ◽  
Exponential Type ◽  
Initial Value Problem ◽  
Energy Estimate ◽  
Linear Wave ◽  
Well Posedness ◽  
Two Dimensional ◽  
Initial Value ◽  
Linear Wave Equation ◽  
Funct Anal

Abstract We consider the initial value problem for a two-dimensional semi-linear wave equation with exponential type nonlinearity. We obtain global well-posedness in the energy space. We also establish the linearization of bounded energy solutions in the spirit of Gérard [J. Funct. Anal. 141: 60–98, 1996]. The proof uses Moser–Trudinger type inequalities and the energy estimate.

Estimates for the controls of the wave equation with a potential

2018 ◽  
Vol 24 (1) ◽  
pp. 289-309 ◽  
Author(s):  
Sorin Micu ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Space Variable ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Boundary Controls ◽  
The One ◽  
Variable Potential ◽  
Uniformly Bounded

This article studies the L2-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential a = a(x). It is known these controls depend on a and their norms may increase exponentially with ||a||L∞. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal L2−norm controls are uniformly bounded with respect to the potential a, if the initial data have only sufficiently high eigenmodes.

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate

scholarly journals Parameter identification for the linear wave equation with Robin boundary condition

2019 ◽  
Vol 27 (1) ◽  
pp. 25-41
Author(s):  
Valeria Bacchelli ◽  
Dario Pierotti ◽  
Stefano Micheletti ◽  
Simona Perotto
Keyword(s):  
Wave Equation ◽  
Mixed Boundary ◽  
Stability Result ◽  
Interval Length ◽  
Left Endpoint ◽  
Linear Wave ◽  
Reconstruction Procedure ◽  
Element Discretization ◽  
Linear Wave Equation ◽  
Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

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