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.

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

2018 ◽  
Vol 2019 (21) ◽  
pp. 6797-6817
Author(s):  
Benjamin Dodson
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Three Dimensions ◽  
Well Posedness ◽  
Critical Space ◽  
Initial Value ◽  
Long Time ◽  
Well Posed

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s> \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

ON THE CAUCHY PROBLEM IN BESOV SPACES FOR A NON-LINEAR SCHRÖDINGER EQUATION

2000 ◽  
Vol 02 (02) ◽  
pp. 243-254 ◽  
Author(s):  
FABRICE PLANCHON
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Initial Value ◽  
Non Linear ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Similar Solutions ◽  
Well Posed

We prove that the initial value problem for a non-linear Schrödinger equation is well-posed in the Besov space [Formula: see text], where the nonlinearity is of type |u|αu. This allows to obtain self-similar solutions, and to recover previous results under weaker smallness assumptions on the data.

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.

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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