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 
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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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.

[1] S. Mandelbroit, Primykayuschie ryady. Regulyarizatsiya posledovatelnostei. Primeneniya, Gostekhizdat, Moskva, 1955

[2] Yu. A. Dubinskii, “O netrivialnosti nekotorykh klassov funktsii i razreshimosti nelineinykh obyknovennykh differentsialnykh uravnenii beskonechnogo poryadka”, Diff. uravneniya, X:2 (1974), 231–240

[3] F. E. Browder, “Nonlinear elliptic boundary value problems”, Bull. Amer. Math. Soc., 69 (1963), 862–874 | DOI | MR | Zbl

[4] G. J. Minty, “On a “monotonicity” method for the solution of nonlinear equations in Banach spaces”, Proc. Nat. Acad. Sci. USA, 50 (1963), 1038–1041 | DOI | MR | Zbl

[5] J. Leray, J. Lions, “Quelques résultats de Višik sur les problemes elliptiques nonlinéaires par les methodes de Minty - Browder”, Bull. Soc. Math. France, 93 (1965), 97–107 | MR | Zbl

[6] Yu. A. Dubinskii, “Kvazilineinye ellipticheskie i parabolicheskie uravneniya lyubogo poryadka”, Uspekhi matem. nauk, XXIII:1 (139) (1968), 45–90 | MR

[7] R. I. Kachurovskii, “Nelineinye monotonnye operatory v banakhovykh prostranstvakh”, Uspekhi matem. nauk, XXIII:2 (140) (1968), 121–168

[8] Zh. Lions, Nekotorye metody reshenii nelineinykh kraevykh zadach, izd-vo «Mir», Moskva, 1972

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