Geodesic
Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation
Sbornik. Mathematics, Tome 27 (1975) no. 2, pp. 143-162 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the study of the Cauchy–Dirichlet problem \begin{gather} L(u)\equiv\sum_{|\alpha|=0}^\infty(-1)^{|\alpha|}D^\alpha A_\alpha(x,\,D^\gamma u)=h(x),\qquad x\in G, \\ D^\omega u\mid_{\partial G}=0,\qquad |\omega|=0,1,\dots, \end{gather} infinite order Sobolev spaces $$ \overset\circ W{}^\infty\{a_\alpha,\,p_\alpha\}\equiv\biggl\{u(x)\in C^\infty_0(G):\rho(u)\equiv\sum^\infty_{|\alpha|=0}a_\alpha\|D^\alpha u\|_{p_\alpha}^{p_\alpha}<\infty\biggr\}, $$ naturally arise, where $a_\alpha\geqslant0$ and $p_\alpha\geqslant1$ are numerical sequences. In this paper criteria for the nontriviality of $\overset\circ W{}^\infty\{a_\alpha,p_\alpha\}$ are established and the problem (1), (2) is investigated. Further, a theorem is obtained on the existence of the limit (as $m\to\infty$) of solutions of nonlinear $2m$th order boundary value problems of elliptic and hyperbolic type, from which, in particular, follows the solvability of the mixed problem for the nonlinear hyperbolic equation $u''+L(u)=h(t,x)$, $t\in[0,T]$, where $T>0$ is arbitrary. Bibliography: 9 titles. 
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Yu. A. Dubinskii. Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation. Sbornik. Mathematics, Tome 27 (1975) no. 2, pp. 143-162. http://geodesic.mathdoc.fr/item/SM_1975_27_2_a0/

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[9] M. M. Vainberg, Variatsionnyi metod i metod monotonnykh operatorov, izd-vo «Nauka», Moskva, 1972 | MR