Structurally damped elastic waves in 2D (Q2830334)

The author studies the equation NEWLINE\[NEWLINE\partial^2_t U- a^2\Delta U-(b^2-a^2)\nabla(\text{div\,}U)+ (-\Delta)^\rho\partial_t U= 0NEWLINE\]NEWLINE proposed as a model for damped elastic waves by \textit{R. C. Charao} et al. [J. Evol. Equ. 14, No. 1, 197--210 (2014; Zbl 1295.35093)]. The positive constants \(a^2\) and \(b^2\) are related to the Lamé constants and \((-\Delta)^\rho\) are fractional powers of the Laplacian, \(0\leq\rho\leq 1\). For \(\rho=0\) we have the classical damping, \(\rho=1\) describes the viscoelastic damping. The author studies the Cauchy problem in the two-dimensional case by Fourier techniques. Very precise results about regularity of solutions are obtained for different values of \(\rho\), in the context of Sobolev and Gevrey spaces.