Method of Galyorkin for nonlinear Sobolev type equations

In this paper the following initial boundary value problem is considered:
$$\left\{
\begin{array}{l} L\left(\frac{\partial u(t,x)}{\partial t}\right)+Mu(t,x)=f(t,x),\\ u(0,x)=u_0(x),\\ D^{\gamma}u\Big|_{\tilde A}=0, |\gamma|<m,\end{array}
\right.$$
$L$ and $M$ are nonlinear differential operators.
It is proved that if $L$ and $M$ satisfy to some conditions, then the sequence constructed by solutions of Galyorkin’s equations for this problem is convergence to the week solution of the problem