scholarly journals Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping

2019 ◽  
Vol 16 (03) ◽  
pp. 495-517 ◽  
Author(s):  
Masahiro Ikeda ◽  
Motohiro Sobajima ◽  
Yuta Wakasugi
Keyword(s):  
Wave Equations ◽  
Time Dependent ◽  
Dimensional Case ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Test Function ◽  
Asymptotic Profile ◽  
Higher Dimensional ◽  
Scaling Variables ◽  
The One

We consider the initial value problem for a semi-linear wave equation with a time-dependent effective damping term. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the coefficient of damping as time tends to infinity. A simple case of effective damping and a threshold case in the sense of overdamping were discussed in Ikeda and Wakasugi (2015) and Ikeda and Inui (2019), respectively. We discuss here general damping terms under a certain assumption that is discussed. The result of this paper is the sharp lifespan estimates of blowup solutions including the typical cases treated in previous studies. The proof of the upper bound of lifespan is a modification of the test-function method by Ikeda–Sobajima (2019) and the one of lower bound is based on the technique of scaling variables introduced by Gallay and Raugel (1998) for the one-dimensional case and Wakasugi (2017) for the higher-dimensional case.

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 $t\to \infty$ .

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

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.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

Restricted orbit equivalence for ergodic ${\Bbb Z}^{d}$ actions I

1997 ◽  
Vol 17 (5) ◽  
pp. 1083-1129 ◽  
Author(s):  
JANET WHALEN KAMMEYER ◽  
DANIEL J. RUDOLPH
Keyword(s):  
Equivalence Relation ◽  
Dimensional Case ◽  
Orbit Equivalence ◽  
One Dimensional ◽  
Ergodic Transformations ◽  
Higher Dimensional ◽  
The One ◽  
Equivalence Invariant ◽  
Classical Entropy

In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equivalence. This is shown to be an equivalence relation. Associated to each size is an entropy which is an equivalence invariant. As in the one-dimensional case this entropy is either the classical entropy or is zero. Several examples are discussed.

CORRELATION AND SPECTRAL PROPERTIES OF MULTIDIMENSIONAL THUE–MORSE SEQUENCES

2007 ◽  
Vol 17 (04) ◽  
pp. 1265-1303 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Spectral Properties ◽  
Three Dimensional ◽  
Number System ◽  
Dimensional Case ◽  
Binary Number ◽  
Singular Continuous Spectrum ◽  
One Dimensional ◽  
Number Systems ◽  
Higher Dimensional ◽  
The One

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in ℤm instead of in ℕ. It is shown that the corresponding ±1-valued Thue–Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.

scholarly journals ON THE NONRELATIVISTIC LIMIT OF LINEAR WAVE EQUATIONS FOR ZERO AND UNITY SPIN PARTICLES

2008 ◽  
Vol 23 (02) ◽  
pp. 129-137 ◽  
Author(s):  
P. YU. MOSHIN ◽  
J. L. TOMAZELLI
Keyword(s):  
Wave Equation ◽  
Power Series ◽  
Unitary Transformation ◽  
Wave Equations ◽  
Power Series Expansion ◽  
Particle Dynamics ◽  
Nonrelativistic Limit ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Linear Wave Equations

The nonrelativistic limit of the linear wave equation for zero and unity spin bosons of mass m in the Duffin–Kemmer–Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy–Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order 1/m2 are taken into account through a recursive iteration procedure.

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