First boundary problem for a nonclassical equation (Q583479)

The following boundary value problem in a cylindrical domain g, \(G=\Omega \times [0,T]\), \(\Omega \subset {\mathbb{R}}^ n\), \(\gamma =\partial \Omega\), \[ Lu\equiv P_{2s+1}(x,t)u+M_{2m}(x,t)u=f(x,t),\quad P_{2s+1}u\equiv \sum^{2s+1}_{i=1}k_ i(x,t)D^ i_ tu, \] \[ M_{2m}u\equiv (-1)^ m\sum_{| \alpha |,| \beta | =m}D^{\alpha}_ x(a_{\alpha \beta}(x,t)D^{\beta}_ xu)+\sum_{| \alpha | \leq 2m-1}a_{\alpha}(x,t)D^{\alpha}_ xu, \] \[ \partial^ ju/\partial n^ j|_{\Gamma}=0,\quad j=0,...,m-1;\quad D^ j_ tu|_{t=0;t=T}=0,\quad j=0,...,s-1;\quad D_ t^ su|_{\bar S_ 0^+}=0,\quad D_ t^ su|_{\bar S_{\bar T}}=0, \] where \(\Gamma =\gamma \times [0,T]\) and \[ S_ 0^{+(-)}=\{(x,0):\quad (-1)^ s k_{2s+1}(x,0)>(<)0\},\quad and\quad S_ T^{+(-)}=\{(x,T):\quad (-1)^ s k_{2s+1}(x,T)>(<)0\} \] is considered. Under certain conditions regarding the coefficients of the equation, the existence of a generalized solution of the above system is proved. Regularity conditions are also given.