An application of L¹ estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

2018/08/08 by Tuan Anh Dao, Dao, Tuan Anh, Michael Reissig +1 · 4 citations
Mathematics · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.1808.02706

Abstract

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models: u_tt+ (-Δ)^σu+ μ(-Δ)^δu_t = f(u,u_t), u(0,x)= u₀(x), u_t(0,x)=u₁(x) with σ≥ 1, μ>0 and δ∈ (0,\fracσ2). Here the function f(u,u_t) stands for the power nonlinearities |u|^p and |u_t|^p with a given number p>1. We are interested in investigating L¹ estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faà di Bruno's formula. By assuming additional L^m regularity on the initial data, we use (L^m∩ L^q)- L^q and L^q- L^q estimates with q∈ (1,∞) and m∈ [1,q), to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on L^q spaces.

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• Existence of global solutions to semilinear damped wave equations with nonlinearities of derivative type
• Structurally damped semilinear evolution equation for positive operators on Hilbert space
• Global in-time rough large data solution to complex-valued semilinear damped evolution equations
• Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities

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