Properties of solutions to the Lamé equations with nonlinear right-hand side (Q1197608)

In this paper a special property of the nonlinear Lamé equation \[ \partial^ 2 u/\partial t^ 2-(\lambda+\mu)\text{grad div}u-\mu\Delta u=f(x,u),\qquad x\in\Omega \tag{1} \] is proved. In (1) \(f=(f_ 1,f_ 2,f_ 3)\) is a given function of \(x=(x_ 1,x_ 2,x_ 3,t)\) and \(u=(u_ 1(x),u_ 2(x),u_ 3(x))\). The constants \(\lambda+2\mu\), \(\mu\) are positive and \(\lambda+2\mu>\mu\), and \(\Omega\) is an open set in \(\mathbb{R}^ 4\). \(u=u(x)\) is assumed to be sufficiently smooth. It is shown that if the solution of (1) \(u\) is radially smooth in the past \((t<0\), \((x_ 1,x_ 2,x_ 3)\in\overline{\Omega}\in\mathbb{R}^ 3)\) then \(u(x)\) belongs to \(C^ \infty(\{x_ 1',x_ 2',x_ 3',t)\): \((x_ 1')^ 2+(x_ 2')^ 2+(x_ 3')^ 2\leq\mu t^ 2\}\cap\Omega\), \(t>0)\).