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- Title
L<sup>1</sup> estimates for oscillating integrals and their applications to semi-linear models with σ-evolution like structural damping.
- Abstract
The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped σ-evolution models: utt + (−∆)σu + μ(−∆)δut = f(u,ut), u(0,x) = u0(x), ut(0,x) = u1(x) with σ ≥ 1, μ > 0 and δ ∈(σ/2,σ]. Our aim is to study two main models including σ-evolution models with structural damping δ ∈(σ/2,σ) and those with visco-elastic damping δ = σ. Here the function f(u,ut) stands for power nonlinearities |u|p and |ut|p with a given number p > 1. We are interested in investigating the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on Lq spaces by assuming additional Lm regularity for the initial data, with q ∈ (1,∞) and m ∈ [1,q)
- Subjects
CAUCHY problem; NONLINEAR theories; SOBOLEV spaces; FUNCTION spaces; MATHEMATICAL models of viscoelasticity
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2019, Vol 39, Issue 9, p5431
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2019222