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

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^2ΔU-\big(b^2-a^2\big)\nabla\text{div } U+(-Δ)^θU_t=F(U),\,\, (t,x)\in[0,\infty)\times\mathbb{R}^3, \end{equation*} where $U=U(t,x)=(U^{(1)}(t,x),U^{(2)}(t,x),U^{(3)}(t,x){)}^{\mathrm{T}}$ 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 $F(U)=(|U^{(3)}{|}^{p_1},|U^{(1)}{|}^{p_2},|U^{(2)}{|}^{p_3}{)}^{\mathrm{T}}$ to get results for the global (in time) existence of small data solutions.

42 pages

Cited by in corpus (5)

• Asymptotic profiles of solutions for regularity-loss type generalized thermoelastic plate equations and their applications
• Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part
• Dissipative structure and diffusion phenomena for doubly dissipative elastic waves in two space dimensions
• On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term
• Decay properties and asymptotic profiles for elastic waves with Kelvin-Voigt damping in 2D