Limit investigation at infinity of the Sobolev-Wiener function classes in tube domains (Q2783086)

The authors investigates generalized solutions of the initial boundary value problem NEWLINE\[NEWLINED^2_t\Delta u+D^2_{x_n}u=0,\;(x,t)\in G=g\times (0, \infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu|_{t=0}= u_1(x),\;D_tu|_{t=0}= u_2(x),\;u|_{ \partial g \times (0,\infty)}=0,NEWLINE\]NEWLINE in a tube domain \(G=g\times (0,\infty)\). It is proved that any generalized solution of the problem belongs to certain function spaces \(\widehat W^{\ell,r}_{p,N}\) called Sobolev-Wiener spaces. A function defined on \(G\) belongs to the function space \(\widehat W^{\ell,r}_{p,N}\) if it belongs to a weighted Sobolev space on \(G\) with weight \((1+t)^{-N}\) and, moreover, the Laplace transforms of some derivatives \(D^k_t u(x,t)\) satisfy certain integral estimates. The behaviour of the derivatives \(D^k_t D^m_xu(x,t)\) as \(t \to \infty\) is investigated. It is proved that the derivatives oscillate in the variable \(t\) and stabilize to zero.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].