About bounded properties of smooth solutions of some differential-operator equations (Q2735866)

It is considered the equation NEWLINE\[NEWLINED_t^\beta u(t,x)+ (-1)^{-[\beta]+1} D_t^{\{\beta\}} A^\alpha u(t,x)=0, \quad (t,x)\in (0,\infty)\times \mathbb{R}^1, \tag{1}NEWLINE\]NEWLINE with the initial condition NEWLINE\[NEWLINED_t^{\{\beta\}} u(t,\cdot)|_{t=0}= f. \tag{2}NEWLINE\]NEWLINE Here \(\beta\in [-3,0)\), \(\alpha> 0\) are fixed numbers, \([\beta]\) is whole party and \(\{\beta\}\) is fractional party of number \(\beta\), \(D_t^\beta\) is operator of fractional differentiation of order \(\beta\) on variable \(t\), \(A\alpha\) is \(\alpha\)-degree of operator \(A\), where NEWLINE\[NEWLINEA_f= \sum_{k=0}^\infty (2k+1)^\nu c_k h_k(x), \quad\nu> 0, \quad f= \sum_{k=0}^\infty c_k h_k(x),NEWLINE\]NEWLINE \(\{h_k \{_{k=0}^\infty\) is orthonormal system of Hermite functions in \(L_2(\mathbb{R}^1)\). If \(\nu= 1\), then operator \(A\) is harmonic oscillator \(-\frac{d^2} {dx^2}+ x^2\), and with \(\alpha= m\in \mathbb{Z}_+\) and \(\beta\) is negative integer the equation (1) is parabolic with increasing coefficients.NEWLINENEWLINETheorem about representation of the solution of (1) is formulated. Also the necessary and sufficient condition are given with which this solution has the limit value (2) in the meaning of generalized functions theory as a functional over some space of basic functions as \(S_\omega^\omega\), which were introduced by I. M. Gelfand and G. E. Shilov in a well-known monograph. There are some misprints.