An initial boundary-value problem for model electromagnetoelasticity system
Let \(Q_T=(-l,l)\times (0,T)\), \(l>0\). Consider a quartuplet of piecewise smooth functions \(r,\nu\): \((-l,l)\to {\mathbb R}\), and \(j,f:Q_T\to {\mathbb R}\) that (may) have jumps at the points \(z = z_k\), \(k=1,\ldots,m\), \(-l<z_1<z_2<\ldots<z_m<l\). Moreover, assume that \(r\) and \(\nu\) are bounded, below and above, by positive numbers. Finally, let \(p\) be a positive number and let \((u_0,u_1)\in H^1_0(-l,l)\times L^2(-l,l)\), \(h_0\in C^\alpha([-l,l])\), \(\alpha\in (0,1)\), with \(h_0(\pm l)=0\). The authors are concerned with the following transmission problem in \(Q_T\) related to the electrodynamics of vibrating elastic media: determine a pair of functions \(h,u:Q_T\to {\mathbb R}\) satisfying the differential system \[ \begin{aligned} h_t& = (rh_z)_z - (hu_t)_z - (rj)_z, \\ u_{tt} &= (\nu^2u_z)_z - phh_z + f, \end{aligned} \] subject to the initial conditions \[ h(z,0)=h_0(z),\quad u(z,0)=u_0(z),\quad u_t(z,0)=u_1(z),\quad z\in (-l,l), \] to the boundary conditions \[ h(-l,t)=h(l,t)=0,\quad u(-l,t)=u(l,t)=0,\quad t\in (0,T), \] and to the transmission conditions \[ [h](z_k,t)=[u](z_k,t)=[r(h_z-j)](z_k,t)=[\nu^2u_z](z_k,t)=0, \] where \([h](z_k,t)=h(z_k+,t)-h(z_k-,t)\), \(k=1,\ldots,m\). Under the previous assumptions on the data the authors show that this transmission problem admits a unique weak solution \((h,u)\) which belongs to the space \(L^\infty((0,T)\); \(H^1(-l,l))\times H^1(Q_T)\) and satisfies the additional property \[ \int_0^{T-\tau}dt\int_{-l}^l | h(z,t+\tau)-h(z,t)| ^2\,dz \to 0\quad \text{ as\;} \tau\to 0. \] Finally, the authors prove the continuous dependence of \((h,u)\) on the data \(f\), \(h_0\), \(u_0\), \(u_1\) and the coefficient \(r\), under the assumptions that such functions, as well as \(\nu\), be smooth on the whole of \(Q_T\).
scientific article; zbMATH DE number 5144840