METHOD OF GALYORKIN FOR NONLINEAR SOBOLEV TYPE EQUATIONS

Abstract

In this paper the following initial boundary value problem is considered: $$ \begin{cases}  L \left(\dfrac{\partial u(t, x)}{\partial t}\right)+Mu (t, x) = f (t, x), \\ u(0, x)=u_0(x), \\ D^{\gamma}u|_{\Gamma}=0, |\gamma|<m,  \end{cases}$$  where $L$ 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.