Large time behavior of solutions to elastic wave with structural damping
In the present paper the author studies the following linear Cauchy problem for elastic wave equations with different structurally damping terms: \begin{align*} & u_{tt}-\mu \Delta u-(\mu+\lambda) \nabla (\operatorname{div} u) + B u_t=0,\,\,\,(t,x)\in (0,\infty)\times \mathbb{R}^n,\\ & \quad u(0,x)=u_0(x), \,\,\,u_t(0,x)=u_1(x),\,\,\, x \in \mathbb{R}^n. \end{align*} Here, \(B u_t\) is either a structurally damped term \(\nu(-\Delta)^\sigma u_t\) or a structurally damped term of Kelvin-Voigt type \(\nu (-\mu \Delta -(\mu+\lambda) \nabla \operatorname{div})^\sigma u_t\) with \(\nu>0\) and \(\sigma \in (0,1]\). The data are supposed to belong to an energy space (even of higher order) with additional \(L^1\) regularity. The main goals of the author are to derive sharp decay estimates of the energies even of higher order and to show the optimality of these estimates by getting estimates from below, too. Moreover, the asymptotic profile of energy solutions even with suitable additional regularity is obtained by using Fourier multipliers connected with classical structurally damped wave equations. The main tools to prove all these results are a careful analysis in the Fourier space, some explicit representations of solutions to classical structurally damped waves, treatment of Fourier multipliers, oscillating integrals, respectively and the theory of special functions. It would be nice to have results (blow-up or global (in time) existence of small data Sobolev solutions) for related semilinear models with power nonlinearities.
scientific article