1,270 research outputs found
A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations
In this paper, we find the critical exponent for global small data solutions to the Cauchy problem in Rn, for dissipative evolution equations with power nonlinearities |u|p or |ut|p,utt+(−Δ)δut+(−Δ)σu=|u|p,|ut|p. Here σ,δ∈N∖0, with 2δ≤σ. We show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation, which are described by the diffusion equations: vt+(−Δ)σ−δv=0,wt+(−Δ)δw=0. The nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method (under suitable sign assumptions on the initial data), and lifespan estimates are obtained. By assuming small initial data in Sobolev spaces, we prove the existence of global solutions in the supercritical case, up to some maximum space dimension n̄, and we derive Lq estimates for the solution, for q∈(1,∞). For σ=2δ, the result holds in any space dimension n≥1. The existence result also remains valid if σ and/or δ are fractional
The Spatio-Temporal Structure of Spiral-Defect Chaos
We present a study of the recently discovered spatially-extended chaotic
state known as spiral-defect chaos, which occurs in low-Prandtl-number,
large-aspect-ratio Rayleigh-Benard convection. We employ the modulus squared of
the space-time Fourier transform of time series of two-dimensional shadowgraph
images to construct the structure factor .
This analysis is used to characterize the average spatial and temporal scales
of the chaotic state. We find that the correlation length and time can be
described by power-law dependences on the reduced Rayleigh number .
These power laws have as yet no theoretical explanation.Comment: RevTex 38 pages with 13 figures. Due to their large size, some
figures are stored as separate gif images. The paper with included hi-res eps
figures (981kb compressed, 3.5Mb uncompressed) is available at
ftp://mobydick.physics.utoronto.ca/pub/MBCA96.tar.gz An mpeg movie and
samples of data are also available at
ftp://mobydick.physics.utoronto.ca/pub/. Paper submitted to Physica
Experiments on wave turbulence : the evolution and growth of second sound acoustic turbulence in superfluid 4He confirm self-similarity.
We report our experiments on the formation of second sound acoustic turbulence in superfluid 4He. The initial growth in spectral amplitude follows power laws that steepen rapidly with increasing harmonic number n, corresponding to a propagating front in frequency space. The lower growth exponents agree well with analytic predictions and numerical modeling. The observed increase in the formation delay with n validates the concept of selfsimilarity in the growth of wave turbulence
An endochronic theory for transversely isotropic fibrous composites
A rational methodology of modelling both nonlinear and elastic dissipative response of transversely isotropic fibrous composites is developed and illustrated with the aid of the observed response of graphite-polyimide off-axis coupons. The methodology is based on the internal variable formalism employed within the text of classical irreversible thermodynamics and entails extension of Valanis' endochronic theory to transversely isotropic media. Applicability of the theory to prediction of various response characteristics of fibrous composites is illustrated by accurately modelling such often observed phenomena as: stiffening reversible behavior along fiber direction; dissipative response in shear and transverse tension characterized by power-laws with different hardening exponents; permanent strain accumulation; nonlinear unloading and reloading; and stress-interaction effects
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