Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation

Yu. A. Dubinskii

Sbornik. Mathematics, Tome 27 (1975) no. 2, pp. 143-162

Résumé

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.