On an infinite order parabolic equation arising in semiconductivity (Q2733866)

This paper deals with the infinite-order parabolic equation NEWLINE\[NEWLINEz_t= z\times \sum_{|\alpha|=0}^\infty a_\alpha D_x^\alpha z+z\times f+B\tag{1}NEWLINE\]NEWLINE with Dirichlet boundary conditions, where \(x\) denotes the vectorial product. The author proves the existence at least one weak solution of (1) under the assumption that the Sobolev space of infinite order NEWLINE\[NEWLINE\overset\circ W_{\{a_\alpha\}}^\infty(\Omega)= \Biggl\{ u\in C_0^\infty(\Omega)\;\Biggl|\;\sum_{|\alpha|=0}^\infty a_\alpha|D^\alpha u|^2< \infty\Biggr\}NEWLINE\]NEWLINE is nontrivial and that \(a_\alpha\) are ``small enough''. NEWLINENEWLINENEWLINENote that in contrast to many cases investigated in the literature, in the case of the author, the estimates obtained in the norm \(\overset \circ W_{\{a_\alpha\}}^\infty (\Omega)\) for the solutions of the \(2M\)th-order problem depend on \(M\) and it is impossible to use the compactness lemma, because \(\overset\circ W_{\{a_\alpha\}}^\infty (\Omega)\) is not reflexive.