A nonlocal boundary-value problem for partial differential equations of infinite order
Consider in a cylinder \(\Omega_ T=\Omega \times(0,T),\) where \(\Omega\) is the tore \([0,2\pi]^ m\), the problem \[ (1)\quad L(\partial /\partial t,\partial /\partial x)u\equiv \sum^{\infty}_{| s| =0}a_ sD^ su=f(x,t),D^ s=\partial^{| s|}/\partial t^{s_ 0}\partial x_{1}^{s_ 1}...\partial x_{m}^{s_ m}, \] \[ (2)\quad \partial^{\alpha}u/\partial t^{\alpha}|_{t=0}-\mu \partial^{\alpha}u/\partial t^{\alpha}|_{t=T}=0\quad(\alpha =0,1,2,...), \] where \(a_ s,\mu \neq 0\) are complex constants. We find the solution of the problem (1),(2) in a Sobolev space of infinite order \(W^{\infty}\{a_ s\}\). In this paper necessary and sufficient conditions for the nontriviality of the spaces \(W^{\infty}\{a_ s\}\) are obtained. The following embedding theorem is proved: Let \(\alpha<(2n-m-1)/4,\) then the space \(W^{\infty}\{a_ s\}\) is embedded into the Sobolev space of finite order \(W^{\alpha}(\Omega_ T)\) for almost all vectors \(\{Re a_ s,Im a_ s\}\quad(s=(0,...^ j..,0,n,0,...,0),\quad j=0,...,m.\) But there exist the spaces \(W^{\alpha}\{a_ s\},\) which are not embedded into \(W^{\alpha}(\Omega_ T).\) It is also proved an existence and uniqueness theorem for the problem (1), (2).
scientific article; zbMATH DE number 3853589