1,720,965 research outputs found
An application of Lp - Lq decay estimates to the semi-linear wave equation with parabolic-like structural damping
No full textA new phenomenon in the critical exponent for structurally damped semi-linear evolution equations
Full text linkIn 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
A classification of structural dissipations for evolution operators
No full textIn this paper, we study the asymptotic profile of the solution for a σ-evolution equation with a time-dependent structural damping. We introduce a classification of the damping term, which clarifies whether the solution behaves like the solution to an anomalous diffusion problem. We call this damping effective, whereas we say that the damping is noneffective when the solution shows oscillations in its asymptotic profile that cannot be neglected. Our classification shows a completely new interplay between the strength of the damping and the long time behavior of its coefficient. Copyright © 2015 John Wiley & Sons, Ltd
The critical exponent for semilinear σ-evolution equations with a strong non-effective damping
No full textIn this paper, we find the critical exponent for the existence of global small data solutions to: [Formula presented]in the case of so-called non-effective damping, θ∈(σ,2σ], where σ≠1 and f=|u|α or f=|ut|α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α>α̃ and do not exist, in general, for subcritical powers 1ᾱ, but we leave open to determine if a counterpart nonexistence result for
Diffusion phenomena for the wave equation with structural damping in the Lp-Lq framework
No full textA class of dissipative wave equations with time-dependent speed and damping
No full textWe study the long time behavior of the energy for wave-type equations with time-dependent speed and damping: utt-λ(t)2δu+b(t)ut=0. We investigate the interaction between the speed of propagationλ (t) and the damping coefficient. b(t), showing how to describe the dissipative effect on the energy. We study a class of dissipations for which the equation keeps its hyperbolic structure and properties. © 2012 Elsevier Ltd
Lp − Lq estimates for a parameter-dependent multiplier with oscillatory and diffusive components
Full text linkIn this paper, we derive long time Lp−Lq decay estimates, in the full range 1≤p≤q≤∞, for time-dependent multipliers in which an interplay between an oscillatory component and a diffusive component with different scaling appears. We estimate ‖m(t,⋅)‖[email protected]@50d4f4b1 as t→∞ for multipliers of type m(t,ξ)=e±i|ξ|javax.xml.bind.JAXBElement@974d21t−|ξ|javax.xml.bind.JAXBElement@3d81aa6bt, and suitable perturbations, under the assumption that the scaling of the diffusive component is worse, i.e., θ>σ. These multipliers are, for instance, related to the fundamental solution to the Cauchy problem for the σ-evolution equation with structural damping: utt+(−Δ)σu+(−Δ)[Formula presented]ut=0,t≥0,x∈Rn, in the so-called non-effective case
Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation
No full textIn this paper we derive asymptotic-in-time linear estimates in Hardy spaces Hp(Rn) for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known Lp- Lq estimates, 1 ≤ p≤ q≤ ∞, for these models. Different, standard, tools to work in Hardy spaces, are used to derive optimal estimates
Hyperbolic-like estimates for higher order equations
No full textAbstractThe main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time-dependent coefficients. In particular, we estimate the energy in the hyperbolic zone of the extended phase space by means of a function f(t) which depends on the principal part and on the coefficients of the terms of order m−1. Then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space. We call this class of estimates hyperbolic-like since the energy behavior is deeply depending on the hyperbolic structure of the equation. In some cases, these estimates produce a dissipative effect on the energy
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