Sufficient conditions for the smoothness of the generalized solution of a nonlocal boundary value problem for a higher-order mixed-type equation (Q5942178)

It is proved that the generalized solution of nonlocal boundary value problem for the equation \[ \sum_{i=1}^{2s} k_i(t,x)D_i^iu-(-1)^m \sum_{|\alpha|=|\beta|=m} D_x^\alpha [a^{\alpha\beta}(x) D_x^\beta u]+ c(t,x)u= f(t,x), \] where \(m\geq 1\), \(s\geq 1\) are integers and \[ \sum_{|\alpha|= |\beta|= m} \xi^\alpha a^{\alpha\beta}(x) \xi^\beta\geq C|\xi|^{2m} \quad\text{for all }\xi\in \mathbb{R}^n, \] where \(C= \text{const}>0\), belongs to the class \(W_{t,x}^{2s-1+l, 2m+(l-1) [m/s]}(G)\), where \(l\geq 1\) is an integer. Sufficient conditions that providing the generalized solution is a classical solution are obtained.