55,792 research outputs found
Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation
We study the focusing, cubic, nonlinear Klein-Gordon equation in 3D with
large radial data in the energy space. This equation admits a unique positive
stationary solution, called the ground state. In 1975, Payne and Sattinger
showed that solutions with energy strictly below that of the ground state are
divided into two classes, depending on a suitable functional: If it is
negative, then one has finite time blowup, if it is nonnegative, global
existence; moreover, these sets are invariant under the flow. Recently,
Ibrahim, Masmoudi and the first author improved this result by establishing
scattering to zero in the global existence case by means of a variant of the
Kenig-Merle method. In this paper we go slightly beyond the ground state energy
and give a complete description of the evolution. For example, in a small
neighborhood of the ground states one encounters the following trichotomy: on
one side of a center-stable manifold one has finite-time blowup, on the other
side scattering to zero, and on the manifold itself one has scattering to the
ground state, all for positive time. In total, the class of initial data is
divided into nine disjoint nonempty sets, each displaying different asymptotic
behavior, which includes solutions blowing up in one time direction and
scattering to zero on the other, and also, the analogue of those found by
Duyckaerts and Merle for the energy critical wave and Schr\"odinger equations,
exactly with the ground state energy. The main technical ingredient is a
"one-pass" theorem which excludes the existence of "almost homoclinic" orbits
between the ground states.Comment: 34 pages, minor correction
New results on group classification of nonlinear diffusion-convection equations
Using a new method and additional (conditional and partial) equivalence
transformations, we performed group classification in a class of variable
coefficient -dimensional nonlinear diffusion-convection equations of the
general form We obtain new interesting cases of
such equations with the density localized in space, which have large
invariance algebra. Exact solutions of these equations are constructed. We also
consider the problem of investigation of the possible local trasformations for
an arbitrary pair of equations from the class under consideration, i.e. of
describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page
Asymptotic dynamics of the exceptional Bianchi cosmologies
In this paper we give, for the first time, a qualitative description of the
asymptotic dynamics of a class of non-tilted spatially homogeneous (SH)
cosmologies, the so-called exceptional Bianchi cosmologies, which are of
Bianchi type VI. This class is of interest for two reasons. Firstly,
it is generic within the class of non-tilted SH cosmologies, being of the same
generality as the models of Bianchi types VIII and IX. Secondly, it is the SH
limit of a generic class of spatially inhomogeneous cosmologies.
Using the orthonormal frame formalism and Hubble-normalized variables, we
show that the exceptional Bianchi cosmologies differ from the non-exceptional
Bianchi cosmologies of type VI in two significant ways. Firstly, the
models exhibit an oscillatory approach to the initial singularity and hence are
not asymptotically self-similar. Secondly, at late times, although the models
are asymptotically self-similar, the future attractor for the vacuum-dominated
models is the so-called Robinson-Trautman SH model instead of the vacuum SH
plane wave models.Comment: 15 pages, 6 figures, submitted to Class. Quantum Gra
Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations
In this article we study global existence and blow-up of solutions for a
general class of nonlocal nonlinear wave equations with power-type
nonlinearities, , where the
nonlocality enters through two pseudo-differential operators and . We
establish thresholds for global existence versus blow-up using the potential
well method which relies essentially on the ideas suggested by Payne and
Sattinger. Our results improve the global existence and blow-up results given
in the literature for the present class of nonlocal nonlinear wave equations
and cover those given for many well-known nonlinear dispersive wave equations
such as the so-called double-dispersion equation and the traditional
Boussinesq-type equations, as special cases.Comment: 17 pages. Accepted for publication in Nonlinear Analysis:Theory,
Methods & Application
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