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arXiv · 1806.08543

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

Wenhui Chen·Michael Reissig
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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 $U=U(t,x)=\big(U^{(1)}(t,x),U^{(2)}(t,x),U^{(3)}(t,x)\big)^{\mathrm{T}}$ with $b^2>a^2>0$ and $θ\in[0,1]$. 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 $θ$ 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.

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Wenhui Chen, Michael Reissig. 2018-09-26. Weakly coupled systems of semi-linear elastic waves with different damping mechanisms in 3D. https://doi.org/10.1002/mma.5370

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