Displaying similar documents to “The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations”

Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation

Rémi Carles, David Lannes (2003)

Bulletin de la Société Mathématique de France

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We consider a system of two linear conservative wave equations, with a nonlinear coupling, in space dimension three. Spherical pulse like initial data cause focusing at the origin in the limit of short wavelength. Because the equations are conservative, the caustic crossing is not trivial, and we analyze it for particular initial data. It turns out that the phase shift between the incoming wave (before the focus) and the outgoing wave (past the focus) behaves like ln ε , where ε stands for...

Bounded Solutions for the Degasperis-Procesi Equation

Giuseppe Maria Coclite, Kenneth H. Karlsen (2008)

Bollettino dell'Unione Matematica Italiana

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This paper deals with the well-posedness in L 1 L of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.

Recent progress in attractors for quintic wave equations

Anton Savostianov, Sergey Zelik (2014)

Mathematica Bohemica

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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...

Double sine series with nonnegative coefficients and Lipschitz classes

Vanda Fülöp (2006)

Colloquium Mathematicae

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Denote by f s s ( x , y ) the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that f s s belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation t u - ³ t x x u + 2 κ x u + x [ g ( u ) / 2 ] = γ ( 2 x u ² x x u + u ³ x x x u ) for the initial data u₀(x) in the Besov space B p , r s ( ) with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given C m -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

A Lipschitz function which is C on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

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We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is C smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...

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

Grzegorz Karch (2000)

Studia Mathematica

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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...

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

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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...

Front propagation for nonlinear diffusion equations on the hyperbolic space

Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)

Journal of the European Mathematical Society

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We study the Cauchy problem in the hyperbolic space n ( n 2 ) for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space n new phenomena arise, which depend on the properties of the diffusion process in n . We also investigate a family of travelling wave solutions,...

Free decay of solutions to wave equations on a curved background

Serge Alinhac (2005)

Bulletin de la Société Mathématique de France

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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.

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz M , definita su un sottoinsieme di uno spazio di Hilbert H a valori compatti e convessi in n , può essere estesa su tutto H ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz M .

Biseparating maps on generalized Lipschitz spaces

Denny H. Leung (2010)

Studia Mathematica

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Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are...

L 2 well-posed Cauchy problems and symmetrizability of first order systems

Guy Métivier (2014)

Journal de l’École polytechnique — Mathématiques

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The Cauchy problem for first order system L ( t , x , t , x ) is known to be well-posed in L 2 when it admits a microlocal symmetrizer S ( t , x , ξ ) which is smooth in ξ and Lipschitz continuous in ( t , x ) . This paper contains three main results. First we show that a Lipschitz smoothness globally in ( t , x , ξ ) is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of having the same smoothness. This notion was first introduced in []. This is the key point to prove...

Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Rivière (2014)

Annales de l’institut Fourier

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We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of β -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of β -damped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic...

Nonlinear Hyperbolic Smoothing at a Focal Point

Jean-Luc Joly, Guy Métivier, Jeffrey Rauch (1998-1999)

Séminaire Équations aux dérivées partielles

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The nonlinear dissipative wave equation u t t - Δ u + | u t | h - 1 u t = 0 in dimension d &gt; 1 has strong solutions with the following structure. In 0 t &lt; 1 the solutions have a focusing wave of singularity on the incoming light cone | x | = 1 - t . In { t 1 } that is after the focusing time, they are smoother than they were in { 0 t &lt; 1 } . The examples are radial and piecewise smooth in { 0 t &lt; 1 }