Dispersive Cases

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Well Posedness ◽  
Decay Properties ◽  
Lp Norms ◽  
Smoothing Effects ◽  
Whole Space ◽  
Simple Systems ◽  
The Waves

It is well known that dispersive phenomena play a significant role in the study of partial differential equations. Historically, the use of dispersive effects appeared in the study of the wave equation in the whole space Rd with the proof of the so-called Strichartz estimates. The idea is that even though the wave equation is time reversible and preserves the energy, it induces a time decay in Lp norms, of course for exponents p greater than 2. In particular, the energy of the waves over a bounded subdomain vanishes as time goes to infinity. These decay properties also yield smoothing effects, which have been the beginning of a long series of works in which the aforementioned smoothing is used in the analysis of nonlinear wave equations to improve the classical well-posedness results. Similar developments have been applied to the non-linear Schrödinger equations. Let us give a flavor of the proof of dispersion estimates in the case of simple systems.

Global well-posedness and self-similarity for semilinear wave equations in a time-weighted framework of Besov type

2020 ◽  
Vol 17 (01) ◽  
pp. 123-139
Author(s):  
Lucas C. F. Ferreira ◽  
Jhean E. Pérez-López
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Besov Spaces ◽  
Wave Equations ◽  
Well Posedness ◽  
Self Similarity ◽  
Semilinear Wave Equations ◽  
Semilinear Wave Equation

We show global-in-time well-posedness and self-similarity for the semilinear wave equation with nonlinearity [Formula: see text] in a time-weighted framework based on the larger family of homogeneous Besov spaces [Formula: see text] for [Formula: see text]. As a consequence, in some cases of the power [Formula: see text], we cover a initial-data class larger than in some previous results. Our approach relies on dispersive-type estimates and a suitable [Formula: see text]-product estimate in Besov spaces.

scholarly journals Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yinghui He ◽  
Yun-Mei Zhao ◽  
Yao Long
Keyword(s):  
Thin Film ◽  
Wave Equation ◽  
Exact Solutions ◽  
Wave Motion ◽  
Wave Equations ◽  
Equation Method ◽  
Nonlinear Wave Equations ◽  
New Exact Solutions ◽  
Simplest Equation Method ◽  
Film Mass Transfer

The simplest equation method presents wide applicability to the handling of nonlinear wave equations. In this paper, we focus on the exact solution of a new nonlinear KdV-like wave equation by means of the simplest equation method, the modified simplest equation method and, the extended simplest equation method. The KdV-like wave equation was derived for solitary waves propagating on an interface (liquid-air) with wave motion induced by a harmonic forcing which is more appropriate for the study of thin film mass transfer. Thus finding the exact solutions of this equation is of great importance and interest. By these three methods, many new exact solutions of this equation are obtained.

scholarly journals Global Attractor for Nonlinear Wave Equations with Critical Exponent on Unbounded Domain

2016 ◽  
Vol 1 (2) ◽  
pp. 581-602 ◽  
Author(s):  
Yuncheng You
Keyword(s):  
Wave Equation ◽  
Global Attractor ◽  
Nonlinear Wave ◽  
Unbounded Domain ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Growth Exponent ◽  
Global Dynamics ◽  
Dissipative Condition

AbstractAsymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝn(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

scholarly journals Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Linear Equations ◽  
Second Order ◽  
Nonlinear Wave Equations ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
Second Order Differential Equations

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

scholarly journals Time-Dependent Wave Equations on Graded Groups

2021 ◽  
Vol 171 (1) ◽  
Author(s):  
Michael Ruzhansky ◽  
Chiara Alba Taranto
Keyword(s):  
Wave Equation ◽  
Lie Groups ◽  
Classical Result ◽  
Wave Equations ◽  
Evolution Equations ◽  
Time Dependent ◽  
Well Posedness ◽  
Local Loss ◽  
Stratified Lie Groups ◽  
Loss Of Regularity

AbstractIn this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$ p -evolution equations for higher order operators on ${{\mathbb{R}}}^{n}$ R n or on groups, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In particular, we describe an interesting local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.

Nonlinear wave equation in an inhomogeneous medium from non-standard singular Lagrangians functional with two occurrences of integrals

2020 ◽  
Vol 21 (7-8) ◽  
pp. 761-766 ◽  
Author(s):  
Rami Ahmad El-Nabulsi
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Inhomogeneous Medium ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Action Functional ◽  
Linear And Nonlinear ◽  
Singular Lagrangians

AbstractIn this communication, we show that a family of partial differential equations such as the linear and nonlinear wave equations propagating in an inhomogeneous medium may be derived if the action functional is replaced by a new functional characterized by two occurrences of integrals where the integrands are non-standard singular Lagrangians. Several features are illustrated accordingly.

scholarly journals Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation

2003 ◽  
Vol 2003 (18) ◽  
pp. 1111-1136 ◽  
Author(s):  
Xiaoping Yuan
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Invariant Manifold ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Invariant Tori ◽  
Nonlinear Wave Equations ◽  
Elliptic Type ◽  
Quasiperiodic Solutions

It is shown that there are plenty of hyperbolic-elliptic invariant tori, thus quasiperiodic solutions for a class of nonlinear wave equations.

The Pettis integration of a perturbed wave equation

1979 ◽  
Vol 86 (1) ◽  
pp. 145-159
Author(s):  
James P. Fink
Keyword(s):  
Wave Equation ◽  
Vector Field ◽  
Nonlinear Wave ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Initial Value

AbstractIn this paper, we investigate the integrability of the vector field of the initial-value problem associated with certain nonlinear wave equations. This vector field involves translations and as such is not a strongly continuous or even strongly measurable L∞-valued function. It is shown that such a vector field, although not generally Pettis integrable, does turn out to be so in an important situation. We then indicate how this result can be used to obtain pseudo-solutions of the initial-value problem.

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