Semilinear structural damped waves

Abstract

We study the Cauchy problem for the semilinear structural damped wave equation with source term uttMathClass-bin−MathClass-rel△uMathClass-bin+μ(MathClass-bin−Δ)σutMathClass-rel=f(u)MathClass-punc,1emquadu(0MathClass-punc,x)MathClass-rel=u0(x)MathClass-punc,1emquadut(0MathClass-punc,x)MathClass-rel=u1(x)MathClass-punc,with σ ∈ (0,1] in space dimension n ≥ 2 and with a positive constant μ. We are interested in the influence of σ on the critical exponent pcrit in | f(u) | ≈ | u | p. This critical exponent is the threshold between global existence in time of small data solutions and blow‐up behavior for some suitable range of p. Our results are optimal for σ = 1 ∕ 2. Copyright © 2013 John Wiley & Sons, Ltd.

Global existence for semi-linear structurally damped σ-evolution models

“…Meanwhile, the authors of [7] proposed a simplified method for the Cauchy problem (4) to obtain L p − L q estimates using only spherical harmonics in the processing of oscillatory integrals. However, due to the simplicity of techniques, the authors assumed the condition n ≥ 2 for obtaining results in the case σ = 1, too.…”

“…The next step was to use other damping mechanisms, for example, structural damping terms of the form (−Δ) δ u t . In the paper [7] the authors discussed Fujita type exponents for semi-linear structurally damped wave models with power non-linearity of the form…”

“…Here we apply (recently proved) results from harmonic analysis (see Appendix A). On the other hand we know from the paper [7], that L p − L q estimates not necessarily on the conjugate line for the corresponding linear models…”

“…The upper bound for n arises from the set of admissible range for p. Smaller p imply that this set is not empty. In the case σ = 1, a = 0 (we do not have a pseudo-differential action) and m = 2 we conclude the assumptions n ∈ [2, 4], p ∈ [2, n n−2 ] and p > 1 + 2 n−1 (see [7]). …”

Asymptotic behavior for abstract evolution differential equations of second order

“…On the other hand, abstract equations like (1.1) are also treated by Ikehata-Nishihara [12], Chill-Haraux [3], Radu-TodorovaYordanov [18] in the case when A 1 = A 2 = I and A 3 = nonnegative self-adjoint operator in a Hilbert space, and by Ikehata-Todorova-Yordanov [13] in the case when A 1 = I and A 2 = A 3 = nonnegative self-adjoint operator in a Hilbert space, in order to investigate the diffusion phenomenon of solutions. In addition to these works, recently in Lu-Reissig [16], IkehataNatsume [11], Charão-da Luz-Ikehata [2], D'Abbicco-Reissig [7] and D'Abbicco-Ebert [5,6] they studied optimal decay estimates of solutions for (1.1) in the case where A 1 = I , A 2 = (− ) θ (θ ∈ [0, 1]), and A 3 = − , which corresponds to the wave equations with structural damping. Quite recently, two papers due to Charão-da Luz-Ikehata [1,10] have been successively published based on a new method introduced in [2].…”