Displaying similar documents to “Decay estimates of solutions of a nonlinearly damped semilinear wave equation”

Decay of correlations for nonuniformly expanding systems

Sébastien Gouëzel (2006)

Bulletin de la Société Mathématique de France

Similarity:

We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in O ( e - c n ) .

Some decay properties for the damped wave equation on the torus

Nalini Anantharaman, Matthieu Léautaud (2012)

Journées Équations aux dérivées partielles

Similarity:

This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012). We address the decay rates of the energy of the damped wave equation when the damping coefficient b does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger...

Wave equation and multiplier estimates on ax + b groups

Detlef Müller, Christoph Thiele (2007)

Studia Mathematica

Similarity:

Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e i t L ψ ( L / λ ) for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive...

Recent progress in attractors for quintic wave equations

Anton Savostianov, Sergey Zelik (2014)

Mathematica Bohemica

Similarity:

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of 3 with damping terms of the form ( - Δ x ) θ t u , where θ = 0 or θ = 1 / 2 . The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when θ = 1 / 2 . For θ = 0 existence of smooth attractors is more complicated...

Free decay of solutions to wave equations on a curved background

Serge Alinhac (2005)

Bulletin de la Société Mathématique de France

Similarity:

We investigate for which metric g (close to the standard metric g 0 ) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (, non integrable) decay conditions on g - g 0 ; in particular, g - g 0 decays like t - 1 2 - ε along wave cones.

Stability in exponential time of Minkowski space-time with a space-like translation symmetry

Cécile Huneau (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in 1 / t of free solutions...

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay, Romain Joly (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We consider the damped wave equation α u t t + u t = u x x - V ' ( u ) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u ( x , t ) = h ( x - s t ) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V . We show that, if the initial data are sufficiently close to the profile of a front for large | x | , the solution of the damped wave equation converges uniformly on to a travelling front as t + . The proof of this...

Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let  P be a long range metric perturbation of the Euclidean Laplacian on  d , d 2 . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to  P . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group e i t f ( P ) where f has a suitable development at zero (resp. infinity). ...

The Landau-Lifshitz equations and the damping parameter

K. Hamdache, M. Tilioua (2006)

Bollettino dell'Unione Matematica Italiana

Similarity:

The present paper is particularly devoted to the damping effect in ferromagnetic materials. We are interested in determining the sensitivity of the LLG method solution to the phenomenological damping parameter a. We discuss the behaviour of the global weak solutions with finite energy of the Landau-Lifshitz equations when the damping parameter a tends either to 0 (underdamped case) or + (overdamped case).

Large data local solutions for the derivative NLS equation

Ioan Bejenaru, Daniel Tataru (2008)

Journal of the European Mathematical Society

Similarity:

We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension n = 2 . Here we prove a similar result for large initial data in all dimensions n 2 .

Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Grzegorz Karch (2000)

Studia Mathematica

Similarity:

Large time behavior of solutions to the generalized damped wave equation u t t + A u t + ν B u + F ( x , t , u , u t , u ) = 0 for ( x , t ) n × [ 0 , ) is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where A u t = u t , B u = - Δ u , and the nonlinear term is either | u t | q - 1 u t or | u | α - 1 u . In this case, the asymptotic profile of solutions...

Dispersive and Strichartz estimates on H-type groups

Martin Del Hierro (2005)

Studia Mathematica

Similarity:

Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as t - p / 2 ) and the Schrödinger equation (decay as t ( 1 - p ) / 2 ), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on N

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We prove the existence of global attractors for the following semilinear degenerate parabolic equation on N : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

Existence and decay in non linear viscoelasticity

Jaime E. Muñoz Rivera, Félix P. Quispe Gómez (2003)

Bollettino dell'Unione Matematica Italiana

Similarity:

In this work we study the existence, uniqueness and decay of solutions to a class of viscoelastic equations in a separable Hilbert space H given by u t t + M ( [ u ] ) A u - 0 t g ( t - τ ) N ( [ u ] ) A u d τ = 0 , in L 2 ( 0 , T ; H ) u ( 0 ) = u 0 , u t ( 0 ) = u 1 where by u t we are denoting [ u ( t ) ] = ( u ( t ) , u t ( t ) , ( A u ( t ) , u t ( t ) ) , A 1 2 u ( t ) 2 , A 1 2 u t ( t ) 2 , A u ( t ) 2 5 A : D A H H is a nonnegative, self-adjoint operator, M , N : R 5 R are C 2 - functions and g : R R is a C 3 -function with appropriates conditions. We show that there exists global solution in time for small initial data. When u t = A 1 2 u 2 and N = 1 , we show the global existence for large initial data u 0 , u 1 taken in the space D A D A 1 / 2 provided they are...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

Similarity:

We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

A sharp Strichartz estimate for the wave equation with data in the energy space

Neal Bez, Keith M. Rogers (2013)

Journal of the European Mathematical Society

Similarity:

We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the L t , x 4 ( 5 + 1 ) norm of the solution in terms of the energy. We also characterise the maximisers.