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.

scholarly journals Time-dependent attractor of wave equations with nonlinear damping and linear memory

2019 ◽  
Vol 17 (1) ◽  
pp. 89-103
Author(s):  
Qiaozhen Ma ◽  
Jing Wang ◽  
Tingting Liu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Nonlinear Damping ◽  
Time Dependent ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time

Abstract In this article, we consider the long-time behavior of solutions for the wave equation with nonlinear damping and linear memory. Within the theory of process on time-dependent spaces, we verify the process is asymptotically compact by using the contractive functions method, and then obtain the existence of the time-dependent attractor in $\begin{array}{} H^{1}_0({\it\Omega})\times L^{2}({\it\Omega})\times L^{2}_{\mu}(\mathbb{R}^{+};H^{1}_0({\it\Omega})) \end{array}$.

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.

A Modified Test Function Method for Damped Wave Equations

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Marcello D’Abbicco ◽  
Sandra Lucente
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Critical Exponent ◽  
Function Method ◽  
Wave Equations ◽  
Time Dependent ◽  
Small Data ◽  
Test Function ◽  
Test Function Method ◽  
Modified Test

AbstractIn this paper we use a modified test function method to derive nonexistence results for the semilinear wave equation with time-dependent speed and damping. The obtained critical exponent is the same exponent of some recent results on global existence of small data solutions.

scholarly journals Scattering for critical wave equations with variable coefficients

2021 ◽  
pp. 1-19
Author(s):  
Shi-Zhuo Looi ◽  
Mihai Tohaneanu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Variable Coefficients ◽  
Energy Decay ◽  
Time Dependent ◽  
Linear Wave ◽  
Decay Estimates ◽  
Local Energy ◽  
Linear Wave Equation ◽  
Semilinear Wave Equation

Abstract We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$ , but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$ .

scholarly journals DIFFERENTIAL FORMS AND WAVE EQUATIONS FOR GENERAL RELATIVITY

1998 ◽  
Vol 07 (06) ◽  
pp. 857-885 ◽  
Author(s):  
STEPHEN R. LAU
Keyword(s):  
General Relativity ◽  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Extrinsic Curvature ◽  
The Cauchy Problem ◽  
Hyperbolic Form ◽  
Tensor Index

In recent papers, Choquet–Bruhat and York and Abrahams, Anderson, Choquet–Bruhat, and York (we refer to both works jointly as AACY) have cast the 3 + 1 evolution equations of general relativity in gauge-covariant and causal "first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's two-form, which in the "time-gauge" is built linearly from the "extrinsic curvature one-form." The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt–Deser–Misner gravitational momentum.

Theory of damped wave models with integrable and decaying in time speed of propagation

2016 ◽  
Vol 13 (02) ◽  
pp. 417-439 ◽  
Author(s):  
Marcelo Rempel Ebert ◽  
Michael Reissig
Keyword(s):  
Cauchy Problem ◽  
Wave Equations ◽  
Propagation Speed ◽  
Time Dependent ◽  
Scale Invariant ◽  
The Cauchy Problem ◽  
Wave Models ◽  
Speed Of Propagation ◽  
Loss Of Regularity

We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is [Formula: see text] We assume [Formula: see text]. Then we propose a classification of dissipation terms in non-effective and effective. In each case we derive estimates for kinetic and elastic type energies by developing a suitable WKB analysis. Moreover, we show optimality of results by the aid of scale-invariant models. Finally, we explain by an example that in some estimates a loss of regularity appears.

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.

Unified error analysis for nonconforming space discretizations of wave-type equations

2018 ◽  
Vol 39 (3) ◽  
pp. 1206-1245 ◽  
Author(s):  
David Hipp ◽  
Marlis Hochbruck ◽  
Christian Stohrer
Keyword(s):  
Wave Equation ◽  
Error Analysis ◽  
Hilbert Spaces ◽  
Error Bounds ◽  
Wave Equations ◽  
Evolution Equations ◽  
Convergence Rates ◽  
Linear Wave ◽  
Element Discretization ◽  
Continuous Space

Abstract This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

scholarly journals The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xudong Luo ◽  
Qiaozhen Ma
Keyword(s):  
Wave Equation ◽  
Existence Of Solution ◽  
Time Dependent ◽  
Well Posedness ◽  
Fractional Damping ◽  
Decay Coefficient ◽  
Forcing Term ◽  
Higher Regularity

<p style='text-indent:20px;'>We investigate the well-posedness and longtime dynamics of fractional damping wave equation whose coefficient <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> depends explicitly on time. First of all, when <inline-formula><tex-math id="M2">\begin{document}$ 1\leq p\leq p^{\ast\ast} = \frac{N+2}{N-2}\; (N\geq3) $\end{document}</tex-math></inline-formula>, we obtain existence of solution for the fractional damping wave equation with time-dependent decay coefficient in <inline-formula><tex-math id="M3">\begin{document}$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $\end{document}</tex-math></inline-formula>. Furthermore, when <inline-formula><tex-math id="M4">\begin{document}$ 1\leq p&lt;p^{*} = \frac{N+4\alpha}{N-2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ u_{t} $\end{document}</tex-math></inline-formula> is proved to be of higher regularity in <inline-formula><tex-math id="M6">\begin{document}$ H^{1-\alpha}\; (t&gt;\tau) $\end{document}</tex-math></inline-formula> and show that the solution is quasi-stable in weaker space <inline-formula><tex-math id="M7">\begin{document}$ H^{1-\alpha}\times H^{-\alpha} $\end{document}</tex-math></inline-formula>. Finally, we get the existence and regularity of time-dependent attractor.</p>

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