Large time behavior of anisotropic electromagnetic/elasticity equations in exterior domains
The authors consider the coupled system of anistotropic electromagnetic elasticity model of the form: \[ \begin{aligned} u_{tt}- \sum_{i,j=1}^3 {\partial\over\partial x_i} \left(A_{ij}(x){\partial u\over\partial x_j}\right)+ u_t+\gamma \operatorname{curl} E=0, \tag{1}\\ \varepsilon(x)E_t- \operatorname{curl} H+\sigma E- \gamma \operatorname{curl} u_t=0, \tag{2}\\ \mu(x)H_t+ \operatorname{curl} E=0, \tag{3}\\ \operatorname{div}(\mu(x)H)=0 \tag{4}\end{aligned} \] in \(\Omega\times(0,\infty)\), \(\Omega\subset\mathbb R^3\) -- is an extension (unbounded with bounded complement and with compact smooth boundary \(\partial\Omega\)), with initial conditions \[ (u,u_t,E,H)|_{t=0}=(u_0,u_1,E_0,H_0)\quad \text{in }\Omega \tag{5} \] and boundary conditions \[ u=0,\quad \eta\times E=0\quad \text{on }\partial\Omega\times(0,\infty) \] where: \(\eta=\eta(x)\) denotes unit normal vector at \(x\in\partial\Omega\), \(u=(u_1,u_2,u_3)\) -- the displacement vector, \(E=(E_1,E_2,E_3)\) the dielectric induction, \(H=(H_1,H_2,H_3)\) -- the magnetic induction, \(\epsilon(x)\), \(\mu(x)\) -- denote the electric permittivity and magnetic permeability \(\sigma>0\), \(\gamma>0\) -- are the coefficients of the electric conductivity and coupling constant \(\eta=\eta(x)\) -- denotes the unit vector at \(x\in\partial\Omega\). The total energy associated with system (1) is given by \[ {\mathcal L}(t)={1\over2} \intop_\Omega\bigg[|u_t(t)|^2+\sum_{i,j}^3A_{ij}(x) {\partial u\over\partial x_j}(t)\cdot{\partial u\over\partial x_i}(t)+ \varepsilon(x)E(t)E(t)+\mu(x)H(t)\cdot H(t)\bigg]\,dx. \] The main result of this paper is laying in the fact that they proved uniform rates of decay of \({\mathcal L}(t)\) as \(t\to\infty\). In the main theorem, the authors prove that \({\mathcal L}(t)\) decays uniformy at the rate \((1+t)^{-1}\). They show that if they choose more regular data, several terms in higher order energy do decay at faster rates. In the proof they use the method of energy estimate.
scientific article; zbMATH DE number 5597523
scientific article