Stabilization of the solutions of an exterior Dirichlet boundary-value problem for the equation of the oscillations of a plate
Let \(\Omega \subset E_ n\) \((n\geq 2)\) be an unbounded domain with compact boundary \(\Gamma\). In the semiinfinite cylinder \((t>0)\times \Omega\) the author considers the equation \[ \frac{\partial^ 2u} {\partial t^ 2}+ (-\Delta)^ pu=0,\qquad p\geq 2, \tag{1} \] with the initial conditions \[ u|_{t=0}=f_ 0(x),\qquad \frac{\partial u}{\partial t}|_{t=0}=f_ 1(x), \tag{2} \] and the boundary conditions \[ \frac{\partial^ ju}{\partial n^ j_ x}|_{x\in \Gamma}=0,\qquad j=0,...,p-1 \text{ if } t>0, \tag{3} \] where \(n_ x\) is normal to the surface \(\Gamma\). If the surface \(\Gamma\) is sufficiently smooth and the functions \(f_ i(x),i=0,1\), are smooth and finite in \(\Omega\) then the author shows that the solution of the exterior Dirichlet boundary-value problem (1)-(3) tends towards zero as \(t\to \infty\), uniformly on each compactum of \({\bar \Omega}\).
exterior Dirichlet boundary-value problem
scientific article; zbMATH DE number 4003654
scientific article