Weakly coupled systems of semi-linear elastic waves with different damping mechanisms in 3D

TL;DR

This paper studies weakly coupled semi-linear elastic wave systems with different damping mechanisms in 3D, analyzing solution properties and the impact of damping on global existence of small data solutions.

Contribution

It investigates the qualitative behavior of solutions to a 3D coupled elastic wave system with various damping, highlighting how damping parameter  influences nonlinear exponents for global solutions.

Findings

Analyzes linear model properties with damping effects.

Identifies conditions for global existence of small solutions.

Shows how damping parameter affects nonlinear exponents.

Abstract

We consider the following Cauchy problem for weakly coupled systems of semi-linear damped elastic waves with a power source non-linearity in three-dimensions: \begin{equation*} U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div } U+(-\Delta)^{\theta}U_t=F(U),\,\, (t,x)\in[0,\infty)\times\mathbb{R}^3, \end{equation*} where $U=U(t,x)=\big(U^{(1)}(t,x),U^{(2)}(t,x),U^{(3)}(t,x)\big)^{\mathrm{T}}$ with $b^2>a^2>0$ and $\theta\in[0,1]$. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right-hand side and the influence of the value of $\theta$ on the exponents $p_1,p_2,p_3$ in $F(U)=\big(|U^{(3)}|^{p_1},|U^{(1)}|^{p_2},|U^{(2)}|^{p_3}\big)^{\mathrm{T}}$ to get results for the global (in time) existence of small data solutions.