Effect of different additional L^m regularity on semi-linear damped σ-evolution models

2021/06/23 by Khaldi Said, Said, Khaldi, Arioui Fatima Zahra +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Black Holes and Theoretical Physics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2106.12286

Abstract

The motivation of the present study is to discuss the global (in time) existence of small data solutions to the following semi-linear structurally damped σ-evolution models: ∂_ttu+(-Δ)^σu+(-Δ)^σ/2∂_tu=|u| ^p, σ≥ 1, pgt;1, where the Cauchy data (u(0,x), ∂_tu(0,x)) will be chosen from energy space on the base of L^q with different additional L^m regularity, namely u(0,x)∈ H^σ,q(ℝⁿ)∩ L^m₁(ℝⁿ) , ∂_tu(0,x)∈ L^q(ℝⁿ)∩ L^m₂(ℝⁿ), q∈(1,∞), m₁, m₂∈ [1,q). Our new results will show that the critical exponent which guarantees the global (in time) existence is really affected by these different additional regularities and will take two different values under some restrictions on m₁, m₂, q, σ and the space dimension n≥1. Moreover, in each case, we have no loss of decay estimates of the unique solution with respect to the corresponding linear models.

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