INITIAL BOUNDARY VALUE PROBLEM FOR SOBOLEV TYPE NONLINEAR EQUATIONS

Abstract

In this paper following initial boundary value problem  is considered: $$\begin{cases}  A\left(\dfrac{\partial u}{\partial t}\right)+Bu=f, \\ u(0)=u_0, \\D^{\gamma}|_{\Gamma}=0, |\gamma|\leq m. \end{cases}$$ Operators $A$ and $B$ are nonlinear and have the following form: $$Au=\displaystyle\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}A_{\alpha}(x, t, D^{\gamma}u), ~~Bu=\displaystyle\sum_{|\alpha|\leq m}(-1)^{|\alpha|}D^{\alpha}B_{\alpha}(x, t, D^{\gamma}u), |\gamma|\leq m. $$

Conditions for functions $A_{\alpha}$ and $B_{\alpha}$ are obtained that lead to existence and uniqueness of solution of the problem in the spaces $L^p(0, T, ^0W^m_p), p\geq 2$.